Sublattice conjecture for c-Cambrian order element 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 $\mathsf{COEP}(c)$ be the set of $c$-Cambrian order element posets.

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

The conjecture was proved in type $A$ and checked computationally for small Coxeter types. The source notes that progress likely requires either the snake-decomposition characterization above or the analogous sublattice conjecture for $\mathsf{COIP}(c)$.