Closure and irreducibility for Grassmannians associated with quivers having parallel paths

Prove that, for an acyclic quiver with parallel paths, the closure of the embedded representation variety in the Grassmannian of submodules constructed from the path quiver overline Q and its commutativity ideal I coincides with the entire Grassmannian, and deduce that the Grassmannian is irreducible.

Background

The paper extends its construction from acyclic quivers without parallel paths to acyclic quivers in which parallel paths are allowed. In this setting, the authors construct a quiver \overline Q whose vertices are paths in the original quiver Q, impose commutativity relations, and define a module M(V_*) together with a Grassmannian X of submodules of prescribed dimension vector.

For every representation V of Q, the construction produces a submodule \overline V of M(V_*), yielding an embedding of the representation variety Rep_d(k\overline Q/I) into X. The authors conjecture that this embedded locus is dense in X. If true, its closure would equal X and would imply irreducibility. The paper verifies irreducibility in one example with Q=1\rightrightarrows2\to3, but does not resolve the assertion in general.