Negative first Hilbert coefficient criterion for Cohen–Macaulayness

Let ${\bf R}$ be a Noetherian local ring that admits an embedding into a big Cohen–Macaulay module, and let $J$ be a parameter ideal.

Negative Hilbert coefficient conjecture.

$$\mathrm{e}_1(J)<0\quad\Longleftrightarrow\quad {\bf R}\text{ is not Cohen--Macaulay}.$$

The claim proposes that, under the stated embedding hypothesis, negativity of the first Hilbert coefficient detects failure of Cohen–Macaulayness. The source presents it as a conjecture and gives no resolution.