Psi-expansion conjecture for interval graphs

Let $G=([n],E)$ be an interval graph. For a composition $\alpha\vDash n$, let $\Psi_\alpha$ be the type 1 power sum quasisymmetric function, let $z_\alpha=z_{\lambda(\alpha)}$, and let $\mathcal{N}_{G,\alpha}$ be the set of permutations defined by requiring that each segment of composition shape $\alpha$ has neither a $G$-descent nor a nontrivial left-to-right $G$-maximum. Let $\omega$ be the involution on $\operatorname{QSym}$. First conjecture.

$$\omega \chi_G[X;q]=\sum_{\alpha\vDash n}\frac{\Psi_\alpha}{z_{\alpha}}\sum_{\sigma \in \mathcal{N}_{G,\alpha}}q^{\widetilde{\mathsf{inv}}_G(\sigma)}.$$

This extends the power-sum expansion known for Dyck graphs and proposes a quasisymmetric analogue for interval graphs.