Intersection property for 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. Intersection-property conjecture. For all $n>0$ and every composition $\alpha$ of $n$, the lattice $\mathcal{T}_{\alpha}$ has the intersection property. Consequently, $\operatorname{CLO}(\mathcal{T}_{\alpha})$ is a meet-semilattice.

The source reports verification by computer for $n\leq 6$. The conjecture would establish a meet-semilattice structure for the core label order in every parabolic Tamari lattice.