  We study spin-2 deformed-AKLT models on the square lattice, specifically a
two-parameter family of $O(2)$-symmetric ground-state wavefunctions as defined
by Niggemann, Kl\"umper, and Zittartz, who found previously that the phase
diagram consists of a N\'eel-ordered phase and a disordered phase which
contains the AKLT point. Using tensor-network methods, we not only confirm the
N\'eel phase but also find an XY phase with quasi-long-range order and a region
adjacent to it, within the AKLT phase, with very large correlation length, and
investigate the consequences of a perfectly factorizable point at the corner of
that phase.
