Frobenius-surjectivity conjecture for degree-zero local cohomology

Let $(R,\mathfrak{m},\mathbb{C})$ be a reduced standard graded $\mathbb{C}$-algebra with an isolated singularity at $\mathfrak{m}$. Let $A$ be a regular subring of $\mathbb{C}$ such that $R$ is defined over $A$. For $\mathfrak{n}\in\operatorname{Max}\operatorname{Spec}(A)$, set $\kappa=A/\mathfrak{n}$ and let $R_\kappa$ and $\mathfrak{m}_\kappa$ denote the corresponding reductions. Frobenius-surjectivity conjecture. The Frobenius map is surjective on $H^i_{\mathfrak{m}_\kappa}(R_\kappa)_0$ for all $i$ and for $\kappa=A/\mathfrak{n}$ in a dense subset of $\mathfrak{n}\in\operatorname{Max}\operatorname{Spec}(A)$. This conjecture proposes a positive-characteristic criterion related to cohomological fullness and the behavior of degree-zero local cohomology under reduction modulo dense sets of maximal ideals; its resolution is not supplied here.