The evenness conjecture for Dynkin quivers

Let $\Gamma$ be a quiver and let $\mathbf{k}$ be a field. Call $\Gamma$ $\mathbf{k}$-even if the complexes $\pi_{y*}\underline{\mathbf{k}}_{\widetilde{F}_{y}}$ are even for every $y\in Y_\nu$, and call $\Gamma$ even if it is $\mathbf{k}$-even for every field $\mathbf{k}$. The evenness conjecture. Each Dynkin quiver is even. The conjecture is motivated by the fact that evenness holds in characteristic zero and no Dynkin quiver was known to fail to be $\mathbf{k}$-even in positive characteristic. A sufficient condition for proving it is that the fibers of the morphisms $\pi_y$ have no odd cohomology over every field.