The j-multiplicity regular-sequence conjecture

Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $M$ be a finitely generated unmixed $R$-module. Let $\mathbf{x}=x_1,\ldots,x_m$ be an amenable partial system of parameters of $M$ that is a $d$-sequence relative to $M$, and let $\mathcal{G}=\mathcal{G}((\mathbf{x}),R)$ be the associated graded ring. Suppose that

$$\dim_{\mathcal{G}}\left(\mathrm{H}_{(\mathbf{x})}(M)\right)=m$$

and that $j_1(\mathbf{x},M)=0$. The j-multiplicity regular-sequence conjecture. Then $\mathbf{x}$ is a regular sequence on $M$. The conjecture proposes that vanishing of the first j-multiplicity under these hypotheses detects regular sequences, analogous to corresponding criteria involving Hilbert coefficients.