Regularity-bounded Stanley decomposition conjecture

Let $S=K[x_1,\ldots,x_n]$ and let $M$ be a finitely generated ${\mathbb Z}^n$-graded $S$-module. A Stanley decomposition of $M$ is a finite direct sum

$$M=\bigoplus_{i=1}^t m_iK[Z_i].$$

Here $m_iK[Z_i]$ denotes a Stanley space, and $|m_i|$ is the degree of its homogeneous generator; let $\operatorname{reg}M$ denote the Castelnuovo–Mumford regularity of $M$. Regularity-bounded Stanley decomposition conjecture. There exists a Stanley decomposition

$$M=\bigoplus_{i=1}^t m_iK[Z_i]$$

of $M$ such that $|m_i|\leq\operatorname{reg}M$ for all $i$. In the squarefree ${\mathbb N}^n$-graded case, the source states that this is equivalent to Stanley's conjecture on Stanley decompositions. The supplied text gives no resolution status.