Discrete configuration-space models for framed cube complexes

Let $K$ be a cube complex such that, if $i\leq k$, every $i$-cube in $K$ is the face of some $k$-cube. Assume moreover that $K$ is framed away from some codimension-$2$ subspace. For a positive integer $n$, let $K(n)$ denote the $n$-fold subdivision of $K$.

Discrete configuration-space conjecture. For all $m\leq n$, the ordered configuration space of $m$ points in $K(n)$ is homotopy equivalent to the $m$th discrete ordered configuration space of $K(n)$.

This conjecture proposes a common discrete model for ordered configuration spaces in sufficiently subdivided pure-dimensional cube complexes with a well-defined frame, extending the surface models discussed in the paper. The supplied text does not state whether the conjecture has been proved or refuted.