Hsu–Logan–Shahriari chain partition conjecture for normalized matching posets

Let $Q$ be a finite graded poset with levels $A_0,\ldots,A_n$. It is rank-symmetric if $|A_i|=|A_{n-i}|$, unimodal if its level sizes increase and then decrease, and a normalized matching poset if it has the LYM property: every antichain $S\subset Q$ satisfies

$$\sum_{i=0}^{n}\frac{|S\cap A_i|}{|A_i|}\leq 1.$$

Let $w$ be the width of $Q$, and set $l=\lfloor |Q|/w\rfloor$.

Hsu–Logan–Shahriari conjecture. If $Q$ is a rank-symmetric, unimodal normalized matching poset of width $w$, then $Q$ can be partitioned into $w$ chains, each chain in the partition having size $l$ or $l+1$.

This extends Füredi's conjecture from the Boolean lattice to rank-symmetric, unimodal normalized matching posets. The source states that this conjecture is open.