Composition-partition invariance conjecture for Dyck graphs

Let $G=([n],E)$ be a Dyck graph. For a composition $\alpha\vDash n$, let $\lambda(\alpha)$ be the partition obtained by rearranging its parts, and let $\mathcal{N}_{G,\alpha}$ be the associated set of permutations. Composition-partition invariance conjecture.

$$\sum_{\sigma \in \mathcal{N}_{G,\alpha}}q^{\widetilde{\mathsf{inv}}_G(\sigma)}=\sum_{\sigma \in \mathcal{N}_{G,\lambda(\alpha)}}q^{\widetilde{\mathsf{inv}}_G(\sigma)}.$$

The paper presents this as equivalent to the preceding Psi-expansion conjecture using the known power-sum formula for Dyck graphs.