The conjecture that virtual specialness is undecidable for non-positively curved cube complexes

Let $X$ be a finite, non-positively curved cube complex. It is virtually special if it has a finite-sheeted covering that is special.

Virtual specialness conjecture. Virtual specialness is an undecidable property of non-positively curved cube complexes.

The preceding theorem proves undecidability for removing self-osculations of a specified hyperplane, but does not settle the recognition problem for virtual specialness, which requires considering all hyperplanes simultaneously. The authors therefore expect a negative solution to the corresponding recognition question.