Given two finite dimensional algebra $A$ and $B$ with generator-cogenerators $M$ and $N$. Define two algebras $X:=End_A(M)$ and $Y:=End_B(N)$. Assume X and Y are derived equivalent. Are A and B derived equivalent?
It is an open problem whether $A$ is selfinjective iff $B$ is selfinjective, see conjecutre 3 of https://link.springer.com/article/10.1007%2Fs11856-016-1327-4.
A positive answers to my quesiton would prove this open problem to be true since derived equivalences preserve being selfinjective. (at least over an algebraically closed field)
edit: A generator is a module having every indecomposable projective module as a direct summand and dually a cogenerator is a module having any injective module as a direct summand. All modules are assumed to be finite dimensional.