Smirnov's stability conjecture for Hilbert-Samuel coefficients of Frobenius powers

Let $(R,\mathfrak m)$ be a Cohen–Macaulay local ring of dimension $d$, and let $I$ be an $\mathfrak m$-primary ideal. The ideal $I$ is stable if it satisfies $I^2=QI$ for a minimal reduction $Q$ of $I$. Smirnov's conjecture. The ideal $I$ is stable if and only if

$$\lim_{q\to\infty}\frac{e_1(I^{[q]})}{q^d}=e_0(I)-e_{HK}(I).$$

The conjecture relates stability of an $\mathfrak m$-primary ideal to the asymptotic first Hilbert–Samuel coefficient of its Frobenius powers and the generalized Hilbert–Kunz multiplicity. It is false whenever $J$ is not a stable ideal, so the stated equivalence is refuted.