Homomorphism conjecture for shard modules of the same dimension

Let $\beta$ be a positive real root, and let $B_1$ and $B_2$ be shard modules of dimension $\beta$. Homomorphism conjecture. Then

$$\operatorname{Hom}(B_1,B_2)\neq 0.$$

This conjecture would imply that the dimension vectors arising from the torsion and torsion-free parts of any torsion pair are disjoint, a necessary condition for associating a biclosed partition of the positive real roots to a torsion pair. Its status is not resolved in the supplied source.