The coprime forest conjecture for friends-and-strangers graphs

Let $X$ and $Y$ be graphs on $n$ vertices. A graph is biconnected if it is connected and has no cut vertex. Write $\overline{Y}$ for the complement of $Y$, and let $\operatorname{FS}(X,Y)$ denote the friends-and-strangers graph associated to $X$ and $Y$. Suppose that

$$\overline{Y}=\mathscr T_1\sqcup\cdots\sqcup\mathscr T_r$$

is a forest whose components are trees, with

$$\gcd\bigl(|V(\mathscr T_1)|,\ldots,|V(\mathscr T_r)|\bigr)=1.$$

Coprime forest conjecture. If $X$ is biconnected, then $\operatorname{FS}(X,Y)$ is connected.

This conjecture asserts that the necessary condition obtained from the cycle case is also sufficient for every biconnected graph $X$. The paper verifies the property for several choices of $Y$, including when the complement is a tree on $n-1$ vertices together with an isolated vertex, but the general assertion is left open.