The biconnected friends-and-strangers connectivity conjecture

Let $Y$ be a graph on $n\geq 3$ vertices such that its complement is a forest consisting of trees $\mathcal T_1,\dots,\mathcal T_r$ with

$$\gcd\bigl(|V(\mathcal T_1)|,\dots,|V(\mathcal T_r)|\bigr)=1.$$

If $X$ is a biconnected graph on $n$ vertices, the biconnected connectivity conjecture asserts that $\mathsf{FS}(X,Y)$ is connected. This generalizes the known characterization for $X=\textsf{Cycle}_n$ to all biconnected graphs and is attributed in the source to Defant et al.; it remains open in the supplied text.