Non-finite generation of Frobenius maps for the determinantal ring

Let $\mathbb{K}$ be a field of prime characteristic $p$, let $R=\mathbb{K}[\![x,y,z,u,v,w]\!]$, and let $I$ be the ideal generated by the $2\times 2$ minors of the matrix

\begin{pmatrix}x&y&z\u&v&w\end{pmatrix}.

Set $S=R/I$, let $E_S$ be the injective hull of the residue field of $S$, and for each $e\geq 1$ let $\mathcal{F}_{<e}$ be the $S$-subalgebra of $\mathcal{F}^e(E_S)$ generated by $\mathcal{F}^1(E_S),\ldots,\mathcal{F}^{e-1}(E_S)$. Non-finite generation conjecture. For all $e\geq 1$, $\mathcal{F}^e(E_S)$ is not contained in $\mathcal{F}_{<e}$ and hence $\mathcal{F}^e(E_S)$ is not a finitely generated $S$-algebra. The ring $S$ is a normal Cohen–Macaulay domain, so this conjecture asks whether the non-finite generation phenomenon occurs over a normal domain. Its resolution would answer the question of whether such examples exist over sufficiently nice rings.