Linear-order conjecture for reduced cycle graphs

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, let $\prec$ be a $\mathfrak{c}$-compatible order of $A_{\mathfrak{c}}$, and let $\Gamma_{\prec}(\mathcal{P}_{g})$ denote the reduced cycle graph associated with the subposet at $g$. Reduced-cycle-graph conjecture. If $\mathcal{P}_{\mathfrak{c}}(G,A)$ is totally chain-connected and admits a $\mathfrak{c}$-compatible order $\prec$, then for every $g\in\mathcal{P}_{\mathfrak{c}}(G,A)$, the reduced cycle graph $\Gamma_{\prec}(\mathcal{P}_{g})$ induces a linear order. By the preceding proposition, this would imply the conjectured total well-coveredness property.