Besicovitch-type product inequality for closed geodesics on higher-dimensional spheres

Let $n>2$, and let $(S^n,g)$ be an $n$-dimensional Riemannian sphere. A Besicovitch-type product inequality asserts that there is a constant $C(n)>0$, depending only on the dimension $n$, such that there exist $n$ distinct closed geodesics with lengths $L_1,\ldots,L_n$ satisfying

$$L_1\cdot L_2\cdot\ldots\cdot L_n\leq C(n)\cdot\operatorname{Vol}(S^n,g).$$

This conjectures a higher-dimensional analogue of the proved two-dimensional inequality, which bounds the product of the lengths of two distinct closed geodesics by the area of a Riemannian $2$-sphere. The higher-dimensional statement remains open in the supplied source.