The finite generation conjecture for finite-dimensional Hopf algebras

Let $A$ be a finite dimensional Hopf algebra over a field $k$, and let $V$ be a finite dimensional $A$-module. Write $H^*(A,k)=\operatorname{Ext}^*_A(k,k)$ for the cohomology algebra of $A$ and $H^*(A,V)=\operatorname{Ext}^*_A(k,V)$ for cohomology with coefficients in $V$. The finite generation conjecture. The cohomology $H^*(A,k)$ is a finitely generated $k$-algebra, and $H^*(A,V)$ is a finitely generated module over $H^*(A,k)$. This is a general finite-generation assertion for Hopf-algebra cohomology; the paper proves it for Drinfeld doubles of Frobenius kernels and related examples, while the assertion for arbitrary finite dimensional Hopf algebras is presented here as a conjecture.