The $j_1$ criterion for regular sequences

Let $M$ be an ${\bf R}$-module satisfying a condition (C), and let $\mathbf{x}=\{x_1,\ldots,x_r\}$ be a partial system of parameters for $M$. Set $\operatorname{gr}_{\mathbf{x}}(M)$ for the associated graded module and let $j_1(\mathbf{x};M)$ be its first generalized Hilbert coefficient.

$j_1$ regular-sequence conjecture. If

$$\dim \mathrm{H}^0_{\mathfrak m}(\operatorname{gr}_{\mathbf{x}}(M))=r \quad\text{and}\quad j_1(\mathbf{x};M)=0,$$

then $\mathbf{x}$ is a regular sequence on $M$. The condition (C) is not fixed in the source; at minimum, $M$ should be unmixed. The source also describes a weak version in which the vanishing is assumed for all partial systems of parameters of $r$ elements. The status is not specified.