Partition conjecture relating characteristic posets and regularity

Let $K$ be a field, $S=K[x_1,\ldots,x_n]$, and let $J\subset I$ be monomial ideals. Let $P_{I/J}$ be the characteristic poset of $I/J$, and let $\operatorname{reg}(I/J)$ denote the regularity of the graded $S$-module $I/J$. For $c,d\in P_{I/J}$, write $[c,d]$ for the interval between them, and for $c=(c_1,\ldots,c_n)$ write $|c|=c_1+\cdots+c_n$. Partition conjecture. There exists a partition

$$\mathcal P: P_{I/J}=\bigcup_{i=1}^r[c_i,d_i]$$

of $P_{I/J}$ such that $|c_i|\leq\operatorname{reg}(I/J)$ for all $i$. The conjecture is presented as following from a conjecture of Soleyman Jahan; the source gives no resolution status.