The sandpile Betti-number conjecture via connected partitions

Let $G=(V,E,s)$ be an undirected sandpile graph. For $U\subseteq V$, let $G|_U$ be the subgraph induced by $U$. A connected $k$-partition of $G$ is a partition $\Pi=\bigsqcup_{i=1}^k V_i$ such that each $G|_{V_i}$ is connected. Let $G_\Pi$ be the graph whose vertices are $V_1,\dots,V_k$, with edge weights

$$\operatorname{wt}(V_i,V_j)=\#\{e\in E:\text{one endpoint of }e\text{ is in }V_i\text{ and the other is in }V_j\}.$$

Treat $G_\Pi$ as a sandpile graph with sink vertex $V_i$ for the unique $i$ such that $s\in V_i$, and let $\mathcal{P}_k$ denote the set of connected $k$-partitions of $G$. The sandpile Betti-number conjecture.

$$\beta_k=\sum_{\Pi\in\mathcal{P}_{k+1}}\#\{c:\text{$c$ a minimal recurrent configuration on }G_\Pi\}.$$

This conjecture identifies the Betti numbers of the relevant sandpile-related algebra with counts of minimal recurrent configurations on partition graphs. It appears as Corollary 3.29 in work of Wilmes and has been verified using Sage for all undirected, unweighted graphs with fewer than $7$ vertices; the general assertion remains open.