Hanes's asymptotic Lech inequality conjecture

Let $(R,\operatorname{\mathfrak{m}})$ be a Noetherian local ring of dimension $d\geq 1$. Write $\widehat{R}$ for its completion, $\operatorname{Spec}\widehat{R}$ for its spectrum, $\operatorname{e}(I)$ for the Hilbert–Samuel multiplicity of an $\operatorname{\mathfrak{m}}$-primary ideal $I$, $\ell(R/I)$ for the length of $R/I$, and $\operatorname{e}(R)=\operatorname{e}(\operatorname{\mathfrak{m}})$. Let $\widehat{R}_{\operatorname{red}}$ denote the reduced quotient of $\widehat{R}$. Hanes's asymptotic Lech inequality conjecture. (a) If $\widehat{R}$ has an isolated singularity, meaning that $\widehat{R}_P$ is regular for every $P\in\operatorname{Spec}\widehat{R}\setminus\{\operatorname{\mathfrak{m}}\}$, then

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

(b) $\operatorname{e}(\widehat{R}_{\operatorname{red}})>1$ if and only if

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

This conjecture proposes an asymptotic refinement of Lech's inequality. Part (b) was subsequently proved in the paper, while the isolated-singularity assertion in part (a) is not resolved by the supplied text.