Shareshian–Wachs's h-unimodality conjecture for Hessenberg representations

Let $P$ be a natural unit interval order on $[n]$. For each $j$, let $\operatorname{ch}H^{2j}({\mathcal H}(P))$ be the Frobenius characteristic of Tymoczko's representation on degree $2j$ cohomology. Shareshian–Wachs's Hessenberg h-unimodality conjecture. The palindromic sequence

$$\bigl(\operatorname{ch}H^{2j}({\mathcal H}(P))\bigr)_{0\leq j\leq |E(\operatorname{inc}(P))|}$$

is $h$-unimodal. The conjecture is proposed as a strengthening of the known Schur-unimodality result of MacPherson and Tymoczko; its general status is open.