Limit conjecture for q-deformations of simplicial-complex arrangements

Let $\Delta=([\ell],\mathcal{F})$ be a simplicial complex on $\ell$ vertices, and let $\mathcal{S}_\Delta^q$ be its generalized $q$-deformation. Let $G$ be the underlying graph of $\Delta$, and write $\chi(\mathcal{S}_\Delta^q,t)$ for the characteristic polynomial of the arrangement and $\chi(G,t)$ for the chromatic polynomial of $G$. Simplicial-complex q-deformation conjecture. The characteristic polynomial $\chi(\mathcal{S}_\Delta^q,t)$ is a polynomial in $q$ and $t$ satisfying

$$\lim_{q\to 1}\frac{\chi(\mathcal{S}_\Delta^q,q^s)}{(q-1)^\ell}=\chi(G,s).$$

This extends the corresponding proposed limit formula for $q$-deformed graphic arrangements from clique complexes to arbitrary simplicial complexes; its resolution is not specified in the supplied text.