Equivariant homology-class conjecture for minimal hypersurfaces

Let $M^{n+1}$ be as in the equivariant setting, with a compact Lie group $G$ acting by isometries and satisfying

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

A $G$-homology class is represented by a fixed $G$-hypersurface $\Sigma_0\subset M$, with representatives differing by the boundary of an open $G$-set. Equivariant homology-class conjecture. $M$ contains infinitely many closed embedded minimal $G$-hypersurfaces $\Sigma$ in a given $G$-homology class, namely

$$\Sigma=\Sigma_0+\partial\Omega$$

as mod $2$ cycles for a fixed $G$-hypersurface $\Sigma_0\subset M$ and some open $G$-set $\Omega\subset M$. This is presented as a stronger equivariant analogue of Yau's conjecture; the context gives no resolution in the stated generality.