(LC) conjecture on Frobenius powers of ideals

Let $(R,\mathfrak{m},k)$ be a local or graded ring of prime characteristic $p>0$, and let $I\subseteq R$ be an ideal, homogeneous in the graded case. Write $I^{[p^e]}$ for the ideal generated by the $p^e$-th powers of elements of $I$, and let $H^0_{\mathfrak{m}}(-)$ denote zeroth local cohomology supported at $\mathfrak{m}$. (LC) conjecture. There exists a positive integer $C$ such that

$$\mathfrak{m}^{Cp^e}\cdot H^0_{\mathfrak{m}}(R/I^{[p^e]})=0$$

for all $e\geq 0$. Establishing this uniform linear bound on the annihilators of the zeroth local cohomology of Frobenius powers would imply, together with weak $F$-regularity, that localizations are $F$-regular; the conjecture is presented as an important application and its resolution is not given in the supplied text.