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Note that the dihedral angle of hawmco at the edge marked by *) below is approximately 133.972805°. Thence the dihedral angle at the base of cubpy, which is exactly 45° still adds to just 178.972805°, which thus is slightly smaller than corealmity with the neighbouring trip. Therefore this augmentation truely remains convex.

Incidence matrix

4 * * * * * | 1 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 2 2 2 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 2 1 2 1 0 0 0 0 0 0 0  augmento-opposite prismatic [3^5]
* 4 * * * * | 0 0 2 0 1 2 1 0 0 0 0 0 0 0 0 0 0 | 0 2 1 0 2 0 2 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 2 0 2 0 0 0 0 0 0  neighbouring prismatic [3^5]
* * 8 * * * | 0 0 0 1 0 1 0 1 1 1 1 0 0 0 0 0 0 | 0 0 0 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 | 0 1 1 1 1 1 1 0 0 0 0  prismatic [3^4,4]
* * * 4 * * | 0 0 0 0 0 0 1 0 0 2 0 1 2 0 0 0 0 | 0 0 0 0 0 0 0 1 2 0 2 0 0 2 2 1 0 0 0 0 0 0 | 0 1 0 0 2 0 2 1 0 0 0  far prismatic [3^5]
* * * * 8 * | 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 | 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 | 0 1 0 0 0 1 1 1 1 1 1  augmented prismatic [3^2,4^2]
* * * * * 1 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 4 4 | 0 0 0 0 0 0 0 0 2 2 2  augmentation tip
------------+-----------------------------------+---------------------------------------------+----------------------
2 0 0 0 0 0 | 2 * * * * * * * * * * * * * * * * | 1 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 2 1 0 0 0 0 0 0 0 0 0  within either base
2 0 0 0 0 0 | * 2 * * * * * * * * * * * * * * * | 1 2 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 2 0 2 1 0 0 0 0 0 0 0  lacing
1 1 0 0 0 0 | * * 8 * * * * * * * * * * * * * * | 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 1 0 0 0 0 0 0 0 0
1 0 1 0 0 0 | * * * 8 * * * * * * * * * * * * * | 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 1 1 1 0 0 0 0 0 0 0
0 2 0 0 0 0 | * * * * 2 * * * * * * * * * * * * | 0 2 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 0 2 0 2 0 0 0 0 0 0  lacing
0 1 1 0 0 0 | * * * * * 8 * * * * * * * * * * * | 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 1 1 0 1 0 0 0 0 0 0
0 1 0 1 0 0 | * * * * * * 4 * * * * * * * * * * | 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 1 0 0 2 0 0 0 0 0 0  meridian within either base
0 0 2 0 0 0 | * * * * * * * 4 * * * * * * * * * | 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 | 0 1 0 1 0 1 0 0 0 0 0  equatorial within either base
0 0 2 0 0 0 | * * * * * * * * 4 * * * * * * * * | 0 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 | 0 0 1 1 1 1 1 0 0 0 0  lacing
0 0 1 1 0 0 | * * * * * * * * * 8 * * * * * * * | 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 | 0 1 0 0 1 0 1 0 0 0 0
0 0 1 0 1 0 | * * * * * * * * * * 8 * * * * * * | 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 | 0 1 0 0 0 1 1 0 0 0 0
0 0 0 2 0 0 | * * * * * * * * * * * 2 * * * * * | 0 0 0 0 0 0 0 1 0 0 2 0 0 0 2 0 0 0 0 0 0 0 | 0 0 0 0 2 0 2 1 0 0 0  lacing
0 0 0 1 1 0 | * * * * * * * * * * * * 8 * * * * | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 | 0 1 0 0 0 0 1 1 0 0 0
0 0 0 0 2 0 | * * * * * * * * * * * * * 4 * * * | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 | 0 1 0 0 0 1 0 0 1 1 0  within either base, orthogonal to augmento-opposite one
0 0 0 0 2 0 | * * * * * * * * * * * * * * 4 * * | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 | 0 1 0 0 0 0 0 1 1 0 1  within either base, parallel to augmento-opposite one *)
0 0 0 0 2 0 | * * * * * * * * * * * * * * * 4 * | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 0 0 1 | 0 0 0 0 0 1 1 1 0 1 1  lacing
0 0 0 0 1 1 | * * * * * * * * * * * * * * * * 8 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 | 0 0 0 0 0 0 0 0 1 1 1  lacing
------------+-----------------------------------+---------------------------------------------+----------------------
4 0 0 0 0 0 | 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 * * * * * * * * * * * * * * * * * * * * * | 2 0 0 0 0 0 0 0 0 0 0  lacing, augmento-opposite
2 2 0 0 0 0 | 0 1 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | * 4 * * * * * * * * * * * * * * * * * * * * | 1 0 1 0 0 0 0 0 0 0 0  lacing
2 1 0 0 0 0 | 1 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | * * 4 * * * * * * * * * * * * * * * * * * * | 1 1 0 0 0 0 0 0 0 0 0
2 0 2 0 0 0 | 0 1 0 2 0 0 0 0 1 0 0 0 0 0 0 0 0 | * * * 4 * * * * * * * * * * * * * * * * * * | 0 0 1 1 0 0 0 0 0 0 0  lacing
1 1 1 0 0 0 | 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 | * * * * 8 * * * * * * * * * * * * * * * * * | 0 1 1 0 0 0 0 0 0 0 0
1 0 2 0 0 0 | 0 0 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 | * * * * * 4 * * * * * * * * * * * * * * * * | 0 1 0 1 0 0 0 0 0 0 0
0 2 2 0 0 0 | 0 0 0 0 1 2 0 0 1 0 0 0 0 0 0 0 0 | * * * * * * 4 * * * * * * * * * * * * * * * | 0 0 1 0 1 0 0 0 0 0 0  lacing
0 2 0 2 0 0 | 0 0 0 0 1 0 2 0 0 0 0 1 0 0 0 0 0 | * * * * * * * 2 * * * * * * * * * * * * * * | 0 0 0 0 2 0 0 0 0 0 0  lacing
0 1 1 1 0 0 | 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 | * * * * * * * * 8 * * * * * * * * * * * * * | 0 1 0 0 1 0 0 0 0 0 0
0 0 4 0 0 0 | 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 | * * * * * * * * * 2 * * * * * * * * * * * * | 0 0 0 1 0 1 0 0 0 0 0  lacing
0 0 2 2 0 0 | 0 0 0 0 0 0 0 0 1 2 0 1 0 0 0 0 0 | * * * * * * * * * * 4 * * * * * * * * * * * | 0 0 0 0 1 0 1 0 0 0 0  lacing
0 0 2 0 2 0 | 0 0 0 0 0 0 0 1 0 0 2 0 0 1 0 0 0 | * * * * * * * * * * * 4 * * * * * * * * * * | 0 1 0 0 0 1 0 0 0 0 0  within either base
0 0 2 0 2 0 | 0 0 0 0 0 0 0 0 1 0 2 0 0 0 0 1 0 | * * * * * * * * * * * * 4 * * * * * * * * * | 0 0 0 0 0 1 1 0 0 0 0  lacing
0 0 1 1 1 0 | 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 | * * * * * * * * * * * * * 8 * * * * * * * * | 0 1 0 0 0 0 1 0 0 0 0
0 0 0 2 2 0 | 0 0 0 0 0 0 0 0 0 0 0 1 2 0 0 1 0 | * * * * * * * * * * * * * * 4 * * * * * * * | 0 0 0 0 0 0 1 1 0 0 0  lacing
0 0 0 1 2 0 | 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 0 0 | * * * * * * * * * * * * * * * 4 * * * * * * | 0 1 0 0 0 0 0 1 0 0 0
0 0 0 0 4 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 0 | * * * * * * * * * * * * * * * * 2 * * * * * | 0 1 0 0 0 0 0 0 1 0 0  within either base
0 0 0 0 4 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 2 0 | * * * * * * * * * * * * * * * * * 2 * * * * | 0 0 0 0 0 1 0 0 0 1 0  lacing, orthogonal to augmento-opposite one
0 0 0 0 4 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 | * * * * * * * * * * * * * * * * * * 2 * * * | 0 0 0 0 0 0 0 1 0 0 1  lacing, parallel to augmento-opposite one
0 0 0 0 2 1 | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 2 | * * * * * * * * * * * * * * * * * * * 4 * * | 0 0 0 0 0 0 0 0 1 1 0
0 0 0 0 2 1 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 | * * * * * * * * * * * * * * * * * * * * 4 * | 0 0 0 0 0 0 0 0 1 0 1
0 0 0 0 2 1 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 | * * * * * * * * * * * * * * * * * * * * * 4 | 0 0 0 0 0 0 0 0 0 1 1
------------+-----------------------------------+---------------------------------------------+----------------------
4 2 0 0 0 0 | 2 2 4 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 2 * * * * * * * * * *  trip
2 2 4 2 4 0 | 1 0 4 4 0 4 2 2 0 4 4 0 4 2 2 0 0 | 0 0 2 0 4 2 0 0 4 0 0 2 0 4 0 2 1 0 0 0 0 0 | * 2 * * * * * * * * *  hawmco
2 2 2 0 0 0 | 0 1 2 2 1 2 0 0 1 0 0 0 0 0 0 0 0 | 0 1 0 1 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | * * 4 * * * * * * * *  trip
2 0 4 0 0 0 | 0 1 0 4 0 0 0 2 2 0 0 0 0 0 0 0 0 | 0 0 0 2 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | * * * 2 * * * * * * *  trip
0 2 2 2 0 0 | 0 0 0 0 1 2 2 0 1 2 0 1 0 0 0 0 0 | 0 0 0 0 0 0 1 1 2 0 1 0 0 0 0 0 0 0 0 0 0 0 | * * * * 4 * * * * * *  trip
0 0 4 0 4 0 | 0 0 0 0 0 0 0 2 2 0 4 0 0 2 0 2 0 | 0 0 0 0 0 0 0 0 0 1 0 2 2 0 0 0 0 1 0 0 0 0 | * * * * * 2 * * * * *  cube
0 0 2 2 2 0 | 0 0 0 0 0 0 0 0 1 2 2 1 2 0 0 1 0 | 0 0 0 0 0 0 0 0 0 0 1 0 1 2 1 0 0 0 0 0 0 0 | * * * * * * 4 * * * *  trip
0 0 0 2 4 0 | 0 0 0 0 0 0 0 0 0 0 0 1 4 0 2 2 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 0 1 0 0 0 | * * * * * * * 2 * * *  trip
0 0 0 0 4 1 | 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 4 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 2 2 0 | * * * * * * * * 2 * *  squippy
0 0 0 0 4 1 | 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 2 4 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 2 | * * * * * * * * * 2 *  squippy
0 0 0 0 4 1 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 4 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 2 | * * * * * * * * * * 2  squippy