Wilmes' multiplicity conjecture for boundary divisors

Let $G=(V,E)$ be an undirected, connected graph. Let $\beta_{k,D}$ be the refined Betti number of a divisor $D$, let $\operatorname{\mathcal{P}}_{k+1}$ be the set of partitions of $G$ into $k+1$ connected components, and let $\operatorname{B}(\Pi)/\operatorname{Img}\operatorname{L}$ denote the relevant boundary-divisor classes modulo the Laplacian image. Suppose that exactly $l$ distinct partitions $\Pi_j$ satisfy the displayed membership condition.

Wilmes' multiplicity conjecture. If there are exactly $l$ distinct partitions $\Pi_j\in\operatorname{\mathcal{P}}_{k+1}$ such that

$$D\in\operatorname{B}(\Pi_j)/\operatorname{Img}\operatorname{L},$$

then

$$\beta_{k,D}=l.$$

This conjecture refines the preceding support characterization by predicting that the refined Betti number records the number of partitions for which the divisor class is a boundary divisor. The source presents it as one of two general expected results; only the top-degree case is established in the supplied text, so the general statement remains open.