Hochster–Huneke's (LC) conjecture

Throughout, let $R:=\bigoplus_{n\geq 0}R_n$ be a standard graded algebra over a field of prime characteristic $p>0$, let $\mathfrak{m}:=\bigoplus_{n>0}R_n$ be its irrelevant ideal, and let $I\lhd R$ be a homogeneous ideal. For $q=p^n$, write $I^{[q]}:=(x^q:x\in I)$ for the $q$-th Frobenius power. Since $\mathfrak{m}^{f(q)}H^0_{\mathfrak{m}}(R/I^{[q]})=0$ for some $f(q)\in\mathbb{N}_0$, the $(LC)$ property concerns linear growth of $f(q)$. Hochster–Huneke's (LC) conjecture. There is some $b\in\mathbb{N}_0$, independent of $q$, such that

$$\mathfrak{m}^{bq}H^0_{\mathfrak{m}}(R/I^{[q]})=0\qquad\forall q.$$

The conjecture predicts a uniform linear bound on the nilpotence exponents of these local-cohomology modules under Frobenius powers. The paper proves it in some non-trivial cases and gives applications, but the general assertion is not resolved by the supplied text.