Equatorial conjecture for equivariant minimal hyperspheres

Let $n\geq4$, let $G=\operatorname{O}(n)\times\operatorname{O}(n)$, and consider $G$-equivariant minimal embeddings of the sphere $S^{2n-1}$ into the round sphere $\mathbb{S}^{2n}$. An embedding is equatorial if its image is an equatorial hypersphere. Equatorial conjecture. For $n\geq4$, any $G$-equivariant, minimal embedding of $S^{2n-1}$ in $\mathbb{S}^{2n}$ is equatorial. The claim is suggested by numerical simulations indicating that, for all tested values $n\geq4$, no non-equatorial trajectory through the centre exits at the relevant boundary. Its status is not resolved in the supplied text.