Minimal-dimension locus conjecture for type A quiver Grassmannians

Let $Q$ be a type $A$ quiver with arbitrary orientation, let $P$ be projective and $I$ injective, and put ${\bf d}=\dim P+\dim I$. For representations $M,N\in{\rm Rep}_{\bf d}$, say that $M$ degenerates to $N$ when $N$ lies in the closure of the orbit of $M$.

Type A minimal-dimension conjecture. There exists a representation $M^2$ of dimension $\dim P$ such that $\Gr_{\dim P}(N)$, for $N\in{\rm Rep}_{\bf d}$, has dimension $\langle\dim P,\dim I\rangle$ if and only if $N$ degenerates to $M^2$.

This asks for a deepest representation controlling the minimal-dimension locus. Special cases have been investigated, but the assertion is presented as conjectural.