The tau-rigidity conjecture for syzygies over dimer tree algebras

Let $B$ be a 2-Calabi–Yau tilted algebra whose quiver satisfies the dimer-tree condition, and let $\tau_B$ denote the Auslander–Reiten translation in $\operatorname{mod}B$. Tau-rigidity conjecture. Every indecomposable syzygy over $B$ is $\tau$-rigid in $\operatorname{mod}B$, namely

$$\operatorname{Hom}_B(M,\tau_B M)=0.$$

The paper notes that indecomposable syzygies over general 2-Calabi–Yau tilted algebras need not be $\tau$-rigid, so the conjecture concerns the special quivers under consideration; the supplied text gives no resolution status.