Irreducible minimal-dimension locus conjecture for principal quiver Grassmannians

Let $Q$ be a Dynkin quiver, let $P$ be projective and $I$ injective, and put ${\bf d}=\dim P+\dim I$. Let ${\rm Rep}_{\bf d}$ be the representation space, and say that $N$ degenerates to $M^1$ if $N$ lies in the closure of the orbit of $M^1$ under the base-change group. Write $\langle\dim P,\dim I\rangle$ for the Euler-form value.

Irreducible-locus conjecture. There exists a representation $M^1$ of dimension $\dim P$ such that, for $N\in{\rm Rep}_{\bf d}$, the quiver Grassmannian $\Gr_{\dim P}(N)$ is irreducible of dimension $\langle\dim P,\dim I\rangle$ if and only if $N$ degenerates to $M^1$. If $\dim P$ and $\dim Q$ have no zero components, then $M^1=P\oplus I$.

Representations degenerating to $P\oplus I$ are known to give the expected dimension, and special equioriented type $A$ cases are known. The general characterization remains conjectural.