PrIncipal quiver Grassmannians: conjectures

Let $P$ and $I$ be a projective and an injective representations of a Dynkin quiver. We consider quiver Grassmannians of subrepresentations of dimension $\dim P$ inside representations of dimension $\dim P+\dim I$. Based on extensive computer experiments, we formulate several conjectures about the algebro-geometric properties of these quiver Grassmannians.