4 problems

Etingof–Ostrik's conjecture. An algebra object in a finite tensor category is exact if and only if it is a finite direct product of simple algebras.

Shimizu's conjecture. Every simple algebra in a finite tensor category is quasi-Frobenius.

Exactness conjecture. Every simple algebra in a finite tensor category is exact, without assuming that the algebra is commutative.

Let $\mathcal{C}$ be the tensor category in the source, and let $A\in\mathcal{C}$ be an indecomposable algebra object. An algebra is called exact when it satisfies the exactness co…