Shareshian–Wachs's power-sum decomposition conjecture II

Let $P$ be a natural unit interval order on $[n]$ with incomparability graph $G$. For a partition $\mu=(\mu_1\geq\cdots\geq\mu_l)\vdash n$, let $z_\mu$ be the standard symmetric-group constant and let $\mathfrak S_{P,\mu,2}$ be the set of $(P,\mu,2)$-compatible permutations defined in the paper. Shareshian–Wachs's second power-sum decomposition conjecture.

$$\omega X_G({\mathbf x},t)=\sum_{\mu\vdash n}z_\mu^{-1}p_\mu\,\prod_{i=1}^{l(\mu)}[\mu_i]_t\sum_{\sigma\in\mathfrak S_{P,\mu,2}}t^{\operatorname{inv}_G(\sigma)}.$$

This is a second proposed refinement of Stanley's power-sum decomposition. The paper presents it as open.