No uniform realisation is possible.

Incidence matrix according to Dynkin symbol

β3x3x3β

both( . . . .    ) | 120 |  1  1  1  1  1 |  1  1  1  1  2  2  1 | 1  1  1 1  2
-------------------+-----+----------------+----------------------+-------------
both( . x . .    ) |   2 | 60  *  *  *  * |  1  1  0  0  1  0  1 | 1  1  0 1  1
both( . . x .    ) |   2 |  * 60  *  *  * |  1  0  1  1  0  1  0 | 1  0  1 1  1
both( s . . s2*a ) |   2 |  *  * 60  *  * |  0  0  0  0  2  2  0 | 0  1  1 0  2
sefa( β3x . .    ) |   2 |  *  *  * 60  * |  1  1  0  1  1  0  0 | 1  1  0 0  1
sefa( . . x3β    ) |   2 |  *  *  *  * 60 |  1  0  1  0  0  1  1 | 0  0  1 1  1
-------------------+-----+----------------+----------------------+-------------
both( . x3x .    ) |   6 |  3  3  0  0  0 | 20  *  *  *  *  *  * | 1  0  0 1  0
      β3x . .      ♦   6 |  3  0  0  3  0 |  * 20  *  *  *  *  * | 1  1  0 0  1
      . . x3β      ♦   6 |  0  3  0  0  3 |  *  * 20  *  *  *  * | 0  0  1 1  0
sefa( β3x3x .    ) |   6 |  0  3  0  3  0 |  *  *  * 20  *  *  * | 1  0  0 0  1
sefa( β3x . β2*a ) |   4 |  1  0  2  1  0 |  *  *  *  * 60  *  * | 0  1  0 0  1
sefa( β . x3β2*a ) |   4 |  0  1  2  0  1 |  *  *  *  *  * 60  * | 0  0  1 0  1
sefa( . x3x3β    ) |   6 |  3  0  0  0  3 |  *  *  *  *  *  * 20 | 0  0  0 1  1
-------------------+-----+----------------+----------------------+-------------
      β3x3x .      ♦  24 | 12 12  0 12  0 |  4  4  0  4  0  0  0 | 5  *  * *  *
      β3x . β2*a   ♦  12 |  6  0  6  6  0 |  0  2  0  0  6  0  0 | * 10  * *  *
      β . x3β2*a   ♦  12 |  0  6  6  0  6 |  0  0  2  0  0  6  0 | *  * 10 *  *
      . x3x3β      ♦  24 | 12 12  0  0 12 |  4  0  4  0  0  0  4 | *  *  * 5  *
sefa( β3x3x3β    ) ♦  12 |  3  3  6  3  3 |  0  0  0  1  3  3  1 | *  *  * * 20

or
both( . . . .    )   | 120 |   2  1   2 |  1  2  2   4 |  2  2  2
---------------------+-----+------------+--------------+---------
both( . x . .    ) & |   2 | 120  *   * |  1  1  1   1 |  2  1  1
both( s . . s2*a )   |   2 |   * 60   * |  0  0  0   4 |  0  2  2
sefa( β3x . .    ) & |   2 |   *  * 120 |  0  1  1   1 |  1  1  1
---------------------+-----+------------+--------------+---------
both( . x3x .    )   |   6 |   6  0   0 | 20  *  *   * |  2  0  0
      β3x . .      & |   6 |   3  0   3 |  * 40  *   * |  1  1  0
sefa( β3x3x .    ) & |   6 |   3  0   3 |  *  * 40   * |  1  0  1
sefa( β3x . β2*a ) & |   4 |   1  2   1 |  *  *  * 120 |  0  1  1
---------------------+-----+------------+--------------+---------
      β3x3x .      & ♦  24 |  24  0  12 |  4  4  4   0 | 10  *  *
      β3x . β2*a   & ♦  12 |   6  6   6 |  0  2  0   6 |  * 20  *
sefa( β3x3x3β    )   ♦  12 |   6  6   6 |  0  0  2   6 |  *  * 20

starting figure: x3x3x3x