Regularity conjecture for amenable d-sequences

Let $(R,\mathfrak m)$ be a Noetherian local ring and $M$ a finite unmixed $R$-module. Let ${\mathbf x}=x_1,\ldots,x_r$ be a partial system of parameters of $M$. The sequence ${\mathbf x}$ is amenable relative to $M$ if the Koszul homology module $\mathrm{H}_1({\mathbf x};M)$ has finite length, and it is a d-sequence relative to $M$ in the sense used in the source. Set

$$\mathrm{G}=\operatorname{gr}_{({\mathbf x})}(R).$$

Regularity conjecture for amenable d-sequences. If ${\mathbf x}$ is an amenable d-sequence relative to $M$,

$$\dim \mathrm{H}^0_{\mathfrak m}(\operatorname{gr}_{({\mathbf x})}(M))=r$$

and $\mathrm{j}_1({\mathbf x};M)=0$, then ${\mathbf x}$ is a regular sequence on $M$.

The claim is a proposed sufficient condition for regularity in the study of partial systems of parameters and Hilbert coefficients of $j$-transforms. The supplied status evidence says that this statement is disproved, so the asserted implication does not hold in general.