F-threshold multiplicity inequality for parameter ideals

Let $(R,\mathfrak{m})$ be a $d$-dimensional Noetherian local ring of characteristic $p>0$. If $J\subseteq\mathfrak{m}$ is an ideal generated by a full system of parameters, and if $\mathfrak{a}\subseteq\mathfrak{m}$ is an $\mathfrak{m}$-primary ideal, define the lower F-threshold

$$\operatorname{c}_{-}^J(\mathfrak{a})=\liminf_{e\to\infty}\frac{\nu^J_{\mathfrak{a}}(p^e)}{p^e},$$

where

$$\nu^J_{\mathfrak{a}}(p^e)=\max\{r\mid\mathfrak{a}^r\not\subseteq J^{[p^e]}\}.$$

Here $J^{[p^e]}$ is generated by the $p^e$-powers of elements of $J$.

F-threshold multiplicity conjecture. One should have

$$e(\mathfrak{a})\geq\left(\frac{d}{\operatorname{c}_{-}^J(\mathfrak{a})}\right)^d e(J).$$

This generalizes characteristic-$p$ multiplicity inequalities involving the F-pure threshold. The case $J=\mathfrak{m}$, where $R$ is regular, was proved by Tucker and Watanabe; the conjecture remains open in general.