The Hilbert–Kunz strengthening of asymptotic Lech's inequality

Let $(R,\operatorname{\mathfrak{m}})$ be a Noetherian local ring of characteristic $p>0$ and dimension $d\geq 1$. Let $\widehat{R}$ be the completion, let $\widehat{R}_{\operatorname{red}}$ be its reduction, let $\operatorname{e}(I)$ denote Hilbert--Samuel multiplicity, let $\ell(R/I)$ denote colength, and let $\operatorname{e}_{\mathrm{HK}}(R)$ denote the Hilbert--Kunz multiplicity of $R$. If $\operatorname{e}(\widehat{R}_{\operatorname{red}})>1$, then

Hilbert–Kunz strengthening.

$$\lim_{N\to\infty}\sup_{\substack{\sqrt{I}=\operatorname{\mathfrak{m}}\ell(R/I)>N}}\left\{\frac{\operatorname{e}(I)}{d!\ell(R/I)}\right\}<\operatorname{e}_{\mathrm{HK}}(R).$$

This strengthens part (2) of the asymptotic Lech conjecture by replacing $\operatorname{e}(R)$ with the potentially smaller Hilbert--Kunz multiplicity. The paper proves the original part (2) in equal characteristic, including cases where $\operatorname{e}_{\mathrm{HK}}(R)=\operatorname{e}(R)$, but the stronger characteristic-$p$ assertion is proposed separately and remains open.