Plücker-dimension characterization of the minimal-dimension locus

Let $Q$ be a Dynkin quiver with arbitrary orientation, let $P$ and $I$ be multiplicity-free projective and injective representations, respectively, and let ${\bf d}=\dim P+\dim I$. For $M\in{\rm Rep}_{\bf d}$, let ${\rm Pl}_{\bf m}(M)$ denote the multihomogeneous component of multidegree ${\bf m}$ in the Plücker algebra of $\Gr_{\dim P}(M)$, and let $M^0$ be a representation in the open orbit.

Plücker-dimension conjecture. For every $M\in{\rm Rep}_{\bf d}$,

$$\dim {\rm Pl}_{\bf m}(M)\geq\dim {\rm Pl}_{\bf m}(M^0)$$

for all ${\bf m}$. Moreover, equality for all ${\bf m}$ holds if and only if $\Gr_{\dim P}(M)$ has dimension $\langle\dim P,\dim I\rangle$.

The conjecture relates graded coordinate-ring dimensions to the geometric minimal-dimension locus. The source gives no resolution and notes that multiplicity assumptions matter.