Classify cube complex groups with all infinite-index subgroups free

Determine whether for every compact, non-positively curved cube complex X in which every subgroup of infinite index in π1(X) is free, the group π1(X) must be either a free group or a surface group.

Background

The main theorem uses separability of hyperplane stabilizers (via virtual specialness) to conclude the existence of one-ended quasiconvex subgroups unless the group is free or a surface group. Removing the virtual specialness/separability assumption would require new methods.

The author notes that combining Theorem A with the flat-closing conjecture would imply a positive answer, but the question remains open without these hypotheses.