Generalized double-flip decomposition conjecture for friends-and-strangers graphs

Let $Y$ be a graph, let $\overline{Y}$ be its complement, and let $\operatorname{Acyc}(\overline{Y})$ denote the acyclic orientations of $\overline{Y}$. Let $\alpha\approx_{ij}\alpha'$ mean that the acyclic orientations are related by a sequence of good $\{i,j\}$-double-flips, and let $\operatorname{Path}_n^{ij}$ be obtained by appending the edge $\{i,j\}$ to $\operatorname{Path}_n$. For each equivalence class $[\alpha]_{\approx_{ij}}$, choose a linear extension $\sigma_{[\alpha]_{\approx_{ij}}}\in\mathcal L([\alpha]_{\approx_{ij}})$, and let $H_{[\alpha]_{\approx_{ij}}}$ be the connected component of $\mathsf{FS}(\operatorname{Path}_n^{ij},Y)$ containing it. Generalized double-flip decomposition conjecture. The component $H_{[\alpha]_{\approx_{ij}}}$ depends only on $[\alpha]_{\approx_{ij}}$, has vertex set $\mathcal L([\alpha]_{\approx_{ij}})$, and

$$\mathsf{FS}(\operatorname{Path}_n^{ij},Y)=\bigoplus_{\alpha\in\operatorname{Acyc}(\overline{Y})/\approx_{ij}}H_{[\alpha]_{\approx_{ij}}}.$$

This is presented as a suspected generalization of an earlier theorem; the paper does not establish it.