Besicovitch-type inequality for a geodesic and minimal hypersurface on higher-dimensional spheres

Let $n>2$, and let $(S^n,g)$ be an $n$-dimensional Riemannian sphere. A Besicovitch-type geodesic–hypersurface inequality asserts that there is a constant $C(n)>0$, depending only on the dimension $n$, such that there exist a closed geodesic $\gamma$ and a smooth closed embedded minimal hypersurface $N^{n-1}$ satisfying

$$\operatorname{Length}(\gamma)\cdot\operatorname{Vol}_{n-1}(N^{n-1})\leq C(n)\cdot\operatorname{Vol}(S^n,g).$$

This is proposed as a second higher-dimensional analogue of the Besicovitch inequality, complementing the conjectured product bound for $n$ distinct closed geodesics. Its resolution status is not established in the supplied source.