Sublattice conjecture for c-Cambrian order interval posets

Let $\Phi$ be a finite root system, let $c$ be a Coxeter element, and let $\mathcal{P}(\Phi)$ be the poset of $\Phi$-posets equipped with its weak order. Let $\mathsf{COIP}(c)$ be the set of $c$-Cambrian order interval posets.

Sublattice conjecture for c-Cambrian order interval posets. For every Coxeter element $c$, the set $\mathsf{COIP}(c)$ induces a sublattice of the weak order on $\mathcal{P}(\Phi)$.

The source explicitly states that this conjecture remains open, although it was proved in type $A$ and verified for small Coxeter types by computation. It is not implied by the established sublattice result inside the weak-order posets because those posets do not themselves form a sublattice of $\mathcal{P}(\Phi)$.