Snake-decomposition characterization of c-Cambrian order element posets

Let $\Phi$ be a finite root system, let $c$ be a Coxeter element, and let $\mathcal{P}(\Phi)$ denote the set of $\Phi$-posets. Let $\mathsf{COEP}(c)$ be the set of $c$-Cambrian order element posets and $\mathsf{COIP}(c)$ the set of $c$-Cambrian order interval posets. A $c$-snake is a sequence of roots satisfying the alternating sign and order conditions described in the source, and a $c$-snake decomposition of a root $\alpha$ is an expression $\alpha=\sum_{i\in[p]}\lambda_i\alpha_i$ with $\lambda_i\in\mathbb{N}$ and $\alpha_1,\ldots,\alpha_p$ a $c$-snake in the poset.

Snake-decomposition characterization. A $\Phi$-poset $\mathrel{\mathop{\rel}}\in\mathcal{P}(\Phi)$ is in $\mathsf{COEP}(c)$ if and only if it is in $\mathsf{COIP}(c)$ and every root $\alpha\in\Phi$ admits a $c$-snake decomposition in $\mathrel{\mathop{\rel}}$.

This characterizes $c$-Cambrian order element posets in terms of $c$-Cambrian order interval posets and snake decompositions. It was proved in type $A$ and checked computationally for small Coxeter types; its general status is not resolved in the supplied source.