Chain-connectedness implies total well-coveredness for compatible orders

Let $G$ be a group generated as a monoid by a conjugation-closed subset $A$, let $\mathfrak{c}\in G$, and let $A_{\mathfrak{c}}$ be the generators appearing in reduced $A$-factorizations of $\mathfrak{c}$. Let $\mathcal{P}_{\mathfrak{c}}(G,A)$ be the factorization poset. A linear order of $A_{\mathfrak{c}}$ is $\mathfrak{c}$-compatible as defined for this poset, and total well-coveredness means that every relevant factorization subposet is well-covered with respect to that order. Well-coveredness conjecture. If $\mathcal{P}_{\mathfrak{c}}(G,A)$ is totally chain-connected and $\prec$ is a $\mathfrak{c}$-compatible order of $A_{\mathfrak{c}}$, then $\mathcal{P}_{\mathfrak{c}}(G,A)$ is totally well-covered with respect to $\prec$. This is stated as an equivalent form of the conjectured EL-labeling criterion.