Homological span conjecture for chromatic homology over \mathcal{A}_m

Let $G$ be a graph with $v$ vertices and $b$ blocks, and let $\mathcal{A}_m=\mathbb{Z}[x]/(x^m=0)$. Write $H_{\mathcal{A}_m}(G)$ for its chromatic homology and $\operatorname{hspan}$ for homological span. Homological span conjecture. The homological span is

$$\operatorname{hspan}(H_{\mathcal{A}_m}(G))=v-b.$$

The source notes that computations indicate invariance under the choice of algebra, while the proved result supplied there gives only the lower bound $\operatorname{hspan}(H_{\mathcal{A}_m}(G))\geq v-b$; equality remains conjectural.