Connectivity conjecture for local dimer dynamics in three dimensions

Let $\mathbf{n}$ be a $3$-dimensional shape, and let $\mathcal{D}_3(\mathbb{Q}_{\mathbf{n}}^3)$ denote the graph of dimer configurations on $\mathbb{Q}_{\mathbf{n}}^3$ connected by local moves of length at most $3$. Connectivity conjecture. For all $3$-dimensional shapes $\mathbf{n}$, the graph $\mathcal{D}_3(\mathbb{Q}_{\mathbf{n}}^3)$ is connected. This conjecture asks whether local dimer dynamics using moves of length at most $3$ can connect every dimer configuration in every three-dimensional shape. It was posed previously by Milet and Saldanha and reiterated by Freire, Klivans, Milet and Saldanha, and by Chandgotia, Sheffield and Wolfram; its status is not established in the supplied source.