Etingof–Ostrik finite generation conjecture for finite tensor categories

Let ${\mathcal C}$ be a finite tensor category, and write

$$\operatorname{H}^*({\mathcal C})=\operatorname{Ext}_{{\mathcal C}}^*(\mathbf{1},\mathbf{1})$$

for its cohomology ring. Etingof–Ostrik finite generation conjecture. The cohomology ring $\operatorname{H}^*({\mathcal C})$ is finitely generated, and $\operatorname{Ext}_{{\mathcal C}}^*(X,X)$ is a finitely generated $\operatorname{H}^*({\mathcal C})$-module for every object $X\in {\mathcal C}$. This conjecture predicts finite generation of cohomology and of the associated self-extension modules for all objects in a finite tensor category; it remains open in the stated generality.