Cycle, barbell, or theta structure conjecture for girth-achieving cycles

Let $X$ be an arbitrary connected graph with finite girth, and let $\mathcal C$ be the set of cycle subgraphs in $\mathsf{FS}(X,\operatorname{Star}_n)$ that achieve its girth. For each $C\in\mathcal C$, let $X_C$ be the path-induced subgraph of $X$ by $C$, and set $\mathcal X=\{X_C\}_{C\in\mathcal C}$. Cycle–barbell–theta structure conjecture. There exists $X_C\in\mathcal X$ isomorphic to a cycle, barbell, or theta graph, and the girth is computed by the corresponding sequence of swaps around that graph. The paper states this as a desired result and supplies only partial structural discussion, leaving the assertion open.