Bayer et al.'s shellability conjecture for cut complexes of grid graphs

Let $\mathcal{G}_{m\times n}$ denote the $m\times n$ grid graph, and let $\Delta_k(G)$ be the $k$-cut complex of a graph $G$. A simplicial complex is shellable if its facets admit a shelling order. Bayer et al.'s shellability conjecture. The complex $\Delta_k(\mathcal{G}_{m\times n})$ is shellable for all

$$3\leq k\leq mn-3.$$

This concerns the shellability of cut complexes across the full stated range of $k$. The source says that Bayer et al. had already proved shellability for $k=3$, while the displayed assertion is presented as their conjecture; no resolution of the full range is given here.