Asymptotic Lech consequence for isolated singularities

Let $(R,\mathfrak{m})$ be a local ring. Suppose its completion $\widehat{R}$ has an isolated singularity, meaning that $\widehat{R}_P$ is regular for every $P\in\operatorname{Spec}(\widehat{R})-\{\mathfrak{m}\}$. If $\operatorname{c_{LM}}(R)>1$, then the supremum in the definition of $\operatorname{c_{LM}}(R)$ is attained. In particular, $\operatorname{c_{LM}}(R)\in\mathbb{Q}$. Asymptotic Lech consequence. The supremum defining $\operatorname{c_{LM}}(R)$ is attained whenever $\operatorname{c_{LM}}(R)>1$ under the isolated-singularity hypothesis above. Without the isolated singularity assumption, this conclusion is false, as shown by an example in the paper.