The simple-curve minimum-girth conjecture for surface subgroups

Let $H$ be a handlebody and let $S$ be a compact surface in $\partial H$. Let $\mathcal{N}(\partial S)$ be the subgroup of $\pi_1(S)$ normally generated by the boundary curves of $S$. Simple-curve minimum-girth conjecture. The minimum girth in $H$ over all closed curves in $\mathcal{N}(\partial S)$ can be achieved by a simple closed curve in $\mathcal{N}(\partial S)$. This extends the type of conclusion sought in the Girth Conjecture and the Main Theorem; the paper notes related positive cases but leaves this formulation for further study.