Flatness and Koszulity conjecture for preprojective algebras of non-ADET graphs

Let $\Gamma$ be a connected graph with adjacency matrix $A$, and let $P({\bf \Gamma})$ be the associated preprojective algebra, constructed from a choice of field $k$ and a modulating space with nondegenerate pairing $E$. The flatness and Koszulity conjecture. (i) The family of preprojective algebras $P({\bf \Gamma})$ is flat: for every choice of $k$ and $E$, its matrix Hilbert series is constant. If $\Gamma$ is not of ADET type, this Hilbert series is

$$(1-At+t^2)^{-1}.$$

(ii) If $\Gamma$ is not of ADET type, then the algebras $P({\bf \Gamma})$ are Koszul.

The claim concerns deformation-invariance of the graded structure of these preprojective algebras and their homological properties outside the ADET cases. The source records no resolution or counterexample, so the conjecture remains open.