The cycle-barbell-theta girth conjecture for star friends-and-strangers graphs

Let $X$ be a connected graph with finite girth, let $\mathcal C$ be the set of cycle subgraphs of $\mathsf{FS}(X,\textsf{Star}_n)$ that achieve its girth, and for each $C\in\mathcal C$ let $X_C$ be the path-induced subgraph of $X$ by $C$. Set

$$\mathcal X=\{X_C\}_{C\in\mathcal C}.$$

The cycle-barbell-theta girth conjecture asserts that some $X_C\in\mathcal X$ is isomorphic to a cycle, barbell, or theta graph, and that the girth can be computed by taking the corresponding sequence of swaps around that graph. The statement is a proposed structural characterization of shortest cycles in these friends-and-strangers graphs; the supplied text gives no resolution.