Chapoton-type identity for parabolic Tamari lattices

Let $n>0$ and let $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_r)$ be a composition of $n$ into $r$ parts. Let $H_{\alpha}(x,y)$ be the polynomial defined on order ideals of the relevant partial order on the transpositions of the parabolic quotient, and let $M_{\alpha}(x,y)$ be the generating function of the Möbius function of the core label order of the parabolic Tamari lattice $\mathcal{T}_{\alpha}$. Chapoton-type identity. The following identity holds if and only if $\alpha$ has at most one part exceeding $1$:

$$H_{\alpha}(x,y)=\Bigl(x(y-1)+1\Bigr)^{r-1}M_{\alpha}\left(\frac{y}{y-1},\frac{x(y-1)}{x(y-1)+1}\right).$$

This conjecture generalizes a connection between Möbius-function generating functions and order-ideal polynomials to parabolic quotients of the symmetric group. The paper reports computational evidence for the stated characterization.