Corrected 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 ideal generated by a full system of parameters of $R$, 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 $J$. Corrected F-threshold and Hilbert-Kunz conjecture.

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

This is proposed as a corrected version after the preceding conjecture is disproved. The supplied text does not establish whether the corrected statement is known or remains open.