The direct-link conjecture for indecomposable modules over ad-quivers

Let $A$ be the path algebra of an ad-quiver $(Q,\sigma)$ without oriented cycles. Let $\beta\in\Delta(Q)^+$ be a positive root, and let $p_F(M)$ and $p_\sigma(\beta)$ denote the periods associated with the Frobenius action on $M$ and the permutation action on $\beta$, respectively. The direct-link conjecture. There exists an indecomposable $A$-module $M$ with dimension vector $\beta$ such that

$$p_F(M)=p_\sigma(\beta).$$

This conjecture is intended to provide a direct link between Kac's theorem and its generalization to modulated quivers. The supplied text does not indicate whether it has been resolved.