Betancort-Smirnov's F-threshold and Hilbert-Kunz conjecture

Let $(R,\mathfrak m)$ be a noetherian local ring of prime characteristic. Let $f_1,f_2,\ldots,f_r$ be part of a system of parameters, let $J$ be an $\mathfrak m$-primary ideal, and set $I=(f_1,\ldots,f_r)R$. Write $e_{HK}$ for Hilbert–Kunz multiplicity and $c^J(I)$ for the $F$-threshold of $I$ with respect to $J$. Betancort-Smirnov's conjecture.

$$e_{HK}(J,R) \leq \left(\frac{c^J(I)}{r}\right)^r e_{HK}\left(J\frac{R}{I},\frac{R}{I}\right).$$

The conjecture was proposed as a comparison between the $F$-threshold and Hilbert–Kunz multiplicity, but the paper gives a proposition disproving it for localizations of polynomial rings in at least two variables over a prime characteristic field.