Free-boundary widths of a solid handlebody

Determine the low free-boundary \(p\)-widths of the mean-convex handlebody \(H_\lambda(C)=\mathbb{S}^3\setminus T_\lambda(C)\) for \(\pi/4\leq\lambda<\pi/2\).

Background

Restricting the canonical family to the complement of a tubular neighborhood of the great circle produces a sweepout of a genus-one handlebody. For the stated range of $\lambda$, the handlebody is mean convex, so free-boundary min–max widths are well posed; their low values remain to be determined.