Stückrad–Vogel conjecture on multiplicity ratios

Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $M$ be a finitely generated $R$-module. Let $e(I,M)$ be the Hilbert–Samuel multiplicity of $M$ with respect to $I$, and set

$$n(M)=\sup_{\sqrt{I+\operatorname{Ann}(M)}=\mathfrak{m}}\left\{\frac{l(M/IM)}{e(I,M)}\right\}.$$

Here, $M$ is quasi-unmixed when its completion $\widehat{M}$ is equidimensional. Stückrad–Vogel's conjecture. One has $n(M)<\infty$ if and only if $M$ is quasi-unmixed.

The conjecture characterizes finiteness of the supremum of length-to-multiplicity ratios in terms of equidimensionality after completion. The paper's abstract states that the upper-bound direction is answered affirmatively, while the source does not establish that the full stated equivalence is solved.