Psi-expansion conjecture for interval-graph permutation functions

Let $G=([n],E)$ be an interval graph and let $\tau\in\mathfrak{S}_n$. Let $\Phi_G(\tau)$, $\mathcal{Q}_{\Phi_G(\tau)}^{(G)}$, $\mathsf{CoInv}_G$, and $\mathsf{Inv}_G$ be the functions and statistics defined in the paper; let $\mathcal{N}_{G,\alpha}$ and $z_\alpha$ be as above, and let $\omega$ be the involution on $\operatorname{QSym}$. General Psi-expansion conjecture.

$$\omega \mathcal{Q}_{\Phi_G(\tau)}^{(G)}=\sum_{\alpha\vDash n}\frac{\Psi_\alpha}{z_{\alpha}}\#\{\sigma\in\mathcal{N}_{G,\alpha}\mid\mathsf{CoInv}_G(\sigma^{-1})=\mathsf{Inv}_G(\tau)\}.$$

This generalizes the preceding interval-graph expansion and relates the involuted quasisymmetric function to permutation statistics.