EL-labeling criterion for factorization posets

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, and let $\lambda_{\mathfrak{c}}$ be its natural labeling. A linear order of $A_{\mathfrak{c}}$ is $\mathfrak{c}$-compatible in the sense defined for the factorization poset. EL-labeling conjecture. The natural labeling $\lambda_{\mathfrak{c}}$ is an EL-labeling of $\mathcal{P}_{\mathfrak{c}}(G,A)$ with respect to some linear order $\prec$ of $A_{\mathfrak{c}}$ if and only if $\mathcal{P}_{\mathfrak{c}}(G,A)$ is totally chain-connected and $\prec$ is $\mathfrak{c}$-compatible. This reformulates the preceding unique-rising-chain conjecture in terms of EL-labelings.