Asymptotic Lech's inequality

Let $(R,\operatorname{\mathfrak{m}})$ be a Noetherian local ring of dimension $d\geq 1$, and let $\widehat{R}$ denote its completion. Write $\operatorname{e}(I)$ for the Hilbert--Samuel multiplicity of an $\operatorname{\mathfrak{m}}$-primary ideal $I$, $\ell(R/I)$ for its colength, and $\operatorname{e}(R)$ for $\operatorname{e}(\operatorname{\mathfrak{m}})$. The completed ring has an isolated singularity when $\widehat{R}_P$ is regular for every $P\in\operatorname{Spec}\widehat{R}-\{\operatorname{\mathfrak{m}}\}$. Then the following should hold:

Asymptotic Lech's inequality.

1. If $\widehat{R}$ has an isolated singularity, then

$$\lim_{N\to\infty}\sup_{\substack{\sqrt{I}=\operatorname{\mathfrak{m}}\ell(R/I)>N}}\left\{\frac{\operatorname{e}(I)}{d!\ell(R/I)}\right\}=1.$$

Equivalently, for every $\varepsilon>0$, there exists $N\gg0$ such that every $\operatorname{\mathfrak{m}}$-primary ideal $I$ with $\ell(R/I)>N$ satisfies

$$\operatorname{e}(I)\leq d!(1+\varepsilon)\ell(R/I).$$

2. One has $\operatorname{e}(\widehat{R}_{\operatorname{red}})>1$ if and only if

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

Equivalently, this holds if and only if there exist $\varepsilon>0$ and $N\gg0$ such that every $\operatorname{\mathfrak{m}}$-primary ideal $I$ with $\ell(R/I)>N$ satisfies

$$\operatorname{e}(I)\leq d!(\operatorname{e}(R)-\varepsilon)\ell(R/I).$$

These assertions seek sharp asymptotic improvements of Lech's inequality for sufficiently deep ideals. Part (1) is known in characteristic $p>0$ when the residue field is perfect, and part (2) is known in equal characteristic; part (1) fails in general without the isolated-singularity hypothesis. The conjecture remains open in the remaining generality.