Huneke–Mustață–Takagi–Watanabe conjecture on F-thresholds and multiplicities

Let $(R,\mathfrak m)$ be a $d$-dimensional Noetherian local ring of characteristic $p>0$. Let $J\subseteq\mathfrak m$ be an ideal generated by a system of parameters, and let $\mathfrak a$ be an $\mathfrak m$-primary ideal. The F-threshold of $\mathfrak a$ with respect to $J$ is

$$\operatorname{c}^{J}(\mathfrak a)=\lim_{e\to\infty}\frac{\min\{N\mid \mathfrak a^N\subseteq(x_1^{p^e},\ldots,x_d^{p^e})\}}{p^e},$$

where $J=(x_1,\ldots,x_d)$. Huneke–Mustață–Takagi–Watanabe conjecture. One has

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

This is the positive-characteristic version of a characteristic-free conjecture of Huneke, Takagi, and Watanabe. The paper proves a special case in dimension two, while the general conjecture remains open.