Tymoczko's Young-subgroup stabilizer conjecture

Let $P$ be a natural unit interval order on $[n]$. Tymoczko's representation of $\mathfrak S_n$ on the cohomology $H^*({\mathcal H}(P))$ is the representation under consideration. Tymoczko's Young-subgroup stabilizer conjecture. For all natural unit interval orders $P$, this representation is a permutation representation in which each point stabilizer is a Young subgroup. The conjecture would, together with the Hessenberg variety conjecture, imply the Stanley–Stembridge conjecture for unit interval orders. The source gives no resolution of the general statement.