Finite flag subcomplex vanishing conjecture

Let $L$ be a finite flag complex, let $\Sigma_L$ be its associated contractible cubical complex, and suppose that $L$ embeds as a full subcomplex of a flag triangulation of the $2k$-sphere.

Finite flag subcomplex vanishing conjecture.

$${\operatorname{\mathcal H}}_i(\Sigma_L)=0\quad\text{for all }i>k.$$

The paper states that this is equivalent to the right-angled Coxeter specialization of Singer's conjecture. It is established only in the ranges obtained from the paper's low-dimensional results, so the general assertion remains open.