Is there an easy classification of piecewise hereditary (which means derived equivalent to a hereditary algebra) algebras of Dynkin type that are Quasi-Frobenius-3 (meaning that the injective envelope of the regular module is projective) by quiver and relations?
Is a piecewise hereditary algebra of Dynkin type $\mathcal{A}$ that is Quasi-Frobenius-3 automatically a Nakayama algebra?
