Reduced path-relation scheme conjecture for principal quiver Grassmannians

Let $Q$ be a Dynkin quiver with arbitrary orientation, let $P$ be a projective representation and $I$ an injective representation, and let $M$ be a representation of dimension vector ${\bf d}=\dim P+\dim I$. The principal quiver Grassmannian is $\Gr_{\dim P}(M)$. For each path in $Q$, impose the associated quadratic Plücker relation.

Reduced scheme conjecture. For every principal quiver Grassmannian, the scheme structure defined by the quadratic relations corresponding to all paths in $Q$ is reduced.

Relations from arrows need not define a reduced scheme structure, so adding relations from all paths is expected to recover the reduced structure. The claim is posed as a conjecture in the paper, with no resolution supplied.