Equivariant Yau conjecture for infinitely many invariant minimal hypersurfaces

Let $M^{n+1}$ be a closed Riemannian manifold equipped with a compact Lie group $G$ acting by isometries, satisfying

$$3 \leq \operatorname{codim}(G\cdot x) \leq 7 \qquad \forall x\in M.$$

A closed embedded minimal hypersurface is $G$-invariant when it is preserved by the action of $G$. Equivariant Yau conjecture. $M$ contains infinitely many closed embedded minimal $G$-hypersurfaces. The assertion is known when $\operatorname{Ric}_M>0$ by equivariant Almgren–Pitts min-max theory, but is open in the stated generality.