Rankedness of the core label order of parabolic Tamari lattices

Let $n>0$ and let $\alpha$ be a composition of $n$. Let $\mathcal{T}_{\alpha}$ denote the parabolic Tamari lattice, and let $\operatorname{CLO}(\mathcal{T}_{\alpha})$ denote its core label order. Rankedness conjecture. For all $n>0$ and every composition $\alpha$ of $n$, the core label order of $\mathcal{T}_{\alpha}$ is ranked.

The poset of parabolic noncrossing partitions is ranked and contains the core label order as a subposet, but the source gives only computational evidence for rankedness of the latter. It also notes that core label orders of congruence-uniform lattices need not be ranked in general.