Acronym  ... 
Name  family of (generally hypercompact) hyperbolic o4x4o2Qo honeycombs (Q≥2) 
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Especially  o4x4o4o (Q=2, paracompact family member) 
This in general hypercompact hyperbolic tesselation uses squats (order 4 square tilings) and also order 2Q square tilings, in the sense of infinite horohedra (i.e. with euclidean curvature) resp. in the sense of infinite bollohedra (i.e. with hyperbolic curvature itself), as its cell types.
For sure, the case Q=2 would be contained here too, resulting in o4x4o4x*a, but this one would be just paracompact (the order 2Q square tilings then would become horohedra too). Moreover it would become regular then.
We might consider for using rational Q=n/d as well. Clearly, in order to not get a Grünbaumian vertex figure, d then has to be odd in general (in the o4x4o4x4*aQ*c setup; for the o4x4o2Qo one rather cf. to o4x4oPo instead). Again Q=n/d>2 here would be hypercompact. And 1<Q=n/d≤2 theoretically would be paracompact. But the subgroup o4o4on/d*a for 1<n/d<2 belongs to spherical geometry and allows just for a single finite realization, cf. the list of Schwarz triangles, where n/d=3/2. But that one would then contradict to our restriction on d.
Incidence matrix according to Dynkin symbol
o4x4o2Qo (Q parametrisable, N,M,K → ∞) . . . .  2NMK  4Q  2Q 4Q  2Q 2 (vertex figure: uniform 2Qprism, scaled by sqrt2) ++++ . x . .  2  4QNMK  1 2  2 1 ++++ o4x . .  4  4  QNMK *  2 0 . x4o .  4  4  * 2QNMK  1 1 ++++ o4x4o .  2M  4M  M M  2QNK * (horohedron with vert. config. 4^4, i.e. squat) . x4o2Qo  2K  2QK  0 QK  * 2NM (bollohedron with vert. config. 4^(2Q)) snubbed forms: o4s4o2Qo
o4x4o4x4*aQ*c (Q parametrisable; N,M,K,L,P → ∞) . . . .  4NMKLP  2Q 2Q  Q Q Q 2Q Q  1 Q 1 Q (vertex figure: uniform 2Qprism, scaled by sqrt2) ++++ . x . .  2  4QNMKLP *  1 0 1 1 0  1 1 0 1 . . . x  2  * 4QNMKLP  0 1 0 1 1  0 1 1 1 ++++ o4x . .  4  4 0  QNMKLP * * * *  1 1 0 0 o . . x4*a  4  0 4  * QNMKLP * * *  0 1 1 0 . x4o .  4  4 0  * * QNMKLP * *  1 0 0 1 . x . x  4  2 2  * * * 2QNMKLP *  0 1 0 1 . . o4x  4  0 4  * * * * QNMKLP  0 0 1 1 ++++ o4x4o . *aQ*c  4M  4QM 0  QM 0 QM 0 0  NKLP * * * (bollohedron with vert. config. 4^(2Q)) o4x . x4*a  4K  4K 4K  K K 0 2K 0  * QNMLP * * (horohedron with vert. config. 4^4, i.e. squat) o . o4x4*aQ*c  4L  0 4QL  0 QL 0 0 QL  * * NMKP * (bollohedron with vert. config. 4^(2Q)) . x4o4x  4P  4P 4P  0 0 P 2P P  * * * QNMKL (horohedron with vert. config. 4^4, i.e. squat) snubbed forms: o4s4o4s4*aQ*c
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