The tiling conjecture for syzygy supports in strongly connected quiver algebras

Let $\Lambda=kQ/I$ be a finite-dimensional algebra whose quiver $Q$ is strongly connected, and let $S_x$ be the simple module associated with a vertex $x$. Write $\Omega^n(S_x)$ for the $n$th syzygy and $\operatorname{supp}\Omega^n(S_x)$ for its vertex support. Tiling conjecture. If

$$\operatorname{pd}S_x=\infty,$$

then $x\in\operatorname{supp}\Omega^n(S_x)$ for infinitely many $n$. This predicts recurrent occurrence of the starting vertex in the syzygy supports of every simple module of infinite projective dimension. The supplied text gives no resolution or known cases for this conjecture.