Homomorphism-dimension criterion for the minimal-dimension locus

Let $Q$ be a Dynkin quiver, let $P$ be projective and $I$ injective, and let $M$ be a representation of dimension vector ${\bf d}=\dim P+\dim I$. Assume either (a) $Q$ is of type $A$ and neither $\dim P$ nor $\dim I$ has zero components, or (b) $Q$ is of type $D$ and $P\oplus I$ contains every indecomposable projective and every indecomposable injective representation as a summand. Let $\operatorname{Hom}_Q(-,-)$ denote the representation homomorphism space.

Homomorphism-dimension conjecture. Under either condition, $\dim\Gr_{\dim P}(M)=\langle\dim P,\dim I\rangle$ if and only if

$$\dim\operatorname{Hom}_Q(M,X)\leq\dim\operatorname{Hom}_Q(P,X)+1$$

for every non-injective indecomposable representation $X$.

This would characterize the minimal-dimension locus by homomorphism dimensions. The source presents it as conjectural, while related special cases and counterexamples outside the stated restrictions are discussed.