The common finite cover conjecture for special cube complexes

Let $X$ and $X'$ be special cube complexes with isomorphic universal covers. A finite-sheeted cover of $X$ and a finite-sheeted cover of $X'$ are said to be common finite covers when they are isomorphic.

Common finite cover conjecture. There exist finite-sheeted covers

$$\hat{X} \longrightarrow X$$

and

$$\hat{X}' \longrightarrow X'$$

such that $\hat{X} \cong \hat{X}'$.

This is a proposed generalization of Leighton's graph covering theorem, and it is intended to provide a route from a common model geometry to commensurability. The context identifies it as a problem first asked by Haglund; its resolution is not established here.