Negative first Hilbert coefficient criterion via faithful big Cohen–Macaulay modules

Let $({\bf R},\mathfrak m)$ be a Noetherian local ring that admits a faithful big Cohen–Macaulay module, and let $J$ be a parameter ideal.

Faithful big Cohen–Macaulay conjecture.

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

This is presented as the expected extension of the corresponding theorem for geometric rings admitting an embedding into a faithful finitely generated maximal Cohen–Macaulay ring. The source gives no resolution.