Polynomial eventual multiplicity conjecture for pure graph braid groups

Let $\mathsf{\Gamma}$ be a graph, let $i$ be a homological degree, and let $\lambda$ be a partition. Write $\mathcal{H}_i(\mathsf{\Gamma})$ for the relevant representation-valued homology object and $\operatorname{mul}_\lambda\mathcal{H}_i(\mathsf{\Gamma})$ for the multiplicity function associated to $\lambda$. Polynomial eventual multiplicity conjecture. For any partition $\lambda$, the function $\operatorname{mul}_\lambda\mathcal{H}_i(\mathsf{\Gamma})$ is eventually equal to a polynomial. This concerns the precise eventual behavior beyond the asymptotic growth results established in the paper. The conjecture remains open.