The q-deformed chromatic polynomial limit conjecture for graphical arrangements

Let $G$ be a graph, let $\mathcal{A}_G^q$ be its $q$-deformed graphical arrangement, let $\chi(\mathcal{A}_G^q,t)$ denote the characteristic polynomial of this arrangement, and let $\chi(G,t)$ denote the chromatic polynomial of $G$. If $\ell$ is the relevant exponent appearing in the normalization, then the q-deformed chromatic polynomial limit conjecture. The function $\chi(\mathcal{A}_G^q,t)$ is a polynomial in $q$ and $t$, and

$$\lim_{q\to 1}\frac{\chi(\mathcal{A}_G^q,q^t)}{(q-1)^\ell}=\chi(G,t).$$

The claim predicts that the ordinary chromatic polynomial is recovered from the $q$-deformed characteristic polynomial by a normalized limit as $q$ approaches $1$. The supplied text presents this as an expectation based on observations for many concrete graphs and does not provide a proof or a precise general specification of $G$ and $\ell$.