Finite-generation converse for Frobenius operators on Q-Gorenstein rings

Let $(R,\mathfrak m)$ be a normal $\mathbb Q$-Gorenstein local ring of prime characteristic, and let $E$ be the injective hull of the residue field $R/\mathfrak m$. The order of the canonical module means its order in the divisor class group.

Finite-generation converse conjecture. If the order of the canonical module is a multiple of the characteristic of $R$, then $\mathcal F(E)$ is not a finitely generated ring extension of $\mathcal F^0(E)$.

This conjecture proposes a converse, in the stated sense, to the result that $\mathcal F(E)$ is finitely generated over $\mathcal F^0(E)$ when the canonical-module order is relatively prime to the characteristic. The paper constructs an example with the asserted failure of finite generation, but the general claim is presented as a conjecture.