Wilmes' conjecture on Betti numbers of graph parking function ideals

Let $G$ be a connected graph with sink vertex $v_n$, let $R=\mathbf{k}[x_1,\ldots,x_{n-1}]$, and let $I$ be the $G$-parking function ideal generated by the monomials associated to connected cuts of $G$. Write $\beta_i(R/I)$ for its coarsely graded Betti numbers, and let $\mathrm{mpf}(\Gamma)$ denote the number of maximal parking functions of a connected graph $\Gamma$. For a connected partition $\pi$ of $G$, write $G\vert_{\pi}$ for the corresponding contracted graph.

Wilmes' conjecture. For all $i\geq 1$, one has

$$\beta_i(R/I)=\sum_{|\pi|=i+1}\mathrm{mpf}(G\vert_{\pi}),$$

where the sum is over all connected partitions $\pi$ of $G$ with $i+1$ parts.

The conjecture predicts the coarsely graded Betti numbers of the graph parking function ideal from maximal parking functions of contracted connected graphs. Wilmes originally formulated the corresponding statement for the topppling ideal, of which $I$ is a distinguished monomial initial ideal; the supplied source gives no evidence of resolution.