Deficiency growth conjecture for Bestvina–Brady approximations

Let $\Sigma$ be a finite flag complex, let $G=G_{\Sigma}$ be the corresponding right-angled Artin group, let $\pi:G\to\mathbf{Z}$ be the canonical surjection, let $H_{\Sigma}=\ker\pi$, and set $\Gamma_n=\pi^{-1}(n\mathbf{Z})$. For a finitely presented group, its deficiency is the maximum, over finite presentations, of the number of generators minus the number of relations.

Deficiency growth conjecture. If $\Sigma$ is not simply connected, then

$$\operatorname{def}(\Gamma_n)\to\infty\quad\text{as }n\to\infty.$$

This is the proposed extension of the paper’s explicit presentation results from one-dimensional flag complexes to arbitrary finite flag complexes. The supplied text does not state whether the conjecture has been resolved.