The bridge characterization conjecture for double-multiplicity friends-and-strangers graphs

Let $X$ be a connected simple graph, and let $c^{(X)}$ and $c^{(\mathsf{Star}_n)}$ be multiplicity lists, with equal total capacity. Let the center of $\mathsf{Star}_n$ have multiplicity

$$k=c^{(\mathsf{Star}_n)}_{\varnothing}.$$

A $k$-bridge is a bridge of the relevant multiplicity determined by the center multiplicity $k$. Let $\operatorname{\mathsf{FS_{m,m}}}(X,\mathsf{Star}_n)$ be the double-multiplicity friends-and-strangers graph. The bridge characterization conjecture. The graph $\operatorname{\mathsf{FS_{m,m}}}(X,\mathsf{Star}_n)$ is connected if and only if $X$ contains no $k$-bridge in which all vertices have multiplicity $1$. This would extend the paper's connectivity characterizations for single-multiplicity friends-and-strangers graphs to the double-multiplicity setting.