Given a finite dimensional algebra $A$ with Jacobson radical $J$. Is the global dimension of $A$ equal to the injective dimension of $J$? I can prove this for algebras with finite global dimension and in general it should be equivalent to the inequality $injdim(J) \geq injdim(J^2)-1$, see my answer.

See also https://mathoverflow.net/questions/288745/injective-dimensions-of-radical-powers .