HMTW's F-threshold and multiplicity conjecture

Let $(R,\mathfrak m)$ be a noetherian local ring of dimension $d$ containing a positive characteristic field. Let $J$ be an ideal generated by a full system of parameters, and let $I$ be an $\mathfrak m$-primary ideal. Here $e(-,R)$ denotes Hilbert–Samuel multiplicity and $c^J(I)$ denotes the $F$-threshold of $I$ with respect to $J$. HMTW's conjecture.

$$e(I,R) \geq \frac{d^d}{c^J(I)^d}e(J,R).$$

The conjecture compares Hilbert–Samuel multiplicity with the $F$-threshold. It is known in the graded case, while the general local statement is not resolved in the supplied text.