Girth-determined torsion-width conjecture over \mathcal{A}_3

Let $G$ be a simple, connected graph with $v$ vertices and girth $\ell$, where $\ell=2k$ or $\ell=2k-1$ according to its parity. Let $P_\ell$ denote the cycle graph of length $\ell$, and let $\mathcal{A}_3=\mathbb{Z}[x]/(x^3=0)$. Write $H_{\mathcal{A}_3}(G)$ for chromatic homology and $\operatorname{hw}^t$ for torsion width. Girth-determined torsion-width conjecture.

$$\operatorname{hw}^t(H_{\mathcal{A}_3}(G))=\operatorname{hw}^t(H_{\mathcal{A}_3}(P_\ell))+v-\ell=(k-1)+v-\ell=\begin{cases}v-k-1,&\ell\text{ even},\v-k,&\ell\text{ odd}. \end{cases}$$

The claim predicts that torsion width over $\mathcal{A}_3$ depends only on the number of vertices and the girth; the source presents it as a conjecture based on computations.