  We show that under suitable hypotheses, the second homotopy group of the
coned-off space associated to a $C(9)$ cubical presentation is trivial, and use
this to provide classifying spaces for proper actions for the fundamental
groups of many quotients of square complexes admitting such cubical
presentations. When the cubical presentations satisfy a condition analogous to
requiring that the relators in a group presentation are not proper powers, we
conclude that the corresponding coned-off space is aspherical.
