Conjecture on infinitely many half-volume constant-mean-curvature hypersurfaces

Let $(M^{n+1},g)$ be a closed Riemannian manifold with $3\le n+1\le 7$. A closed hypersurface encloses half the volume of $M$ when one of the regions it bounds has volume equal to half of $\operatorname{Vol}(M)$. Half-volume CMC conjecture. Every closed Riemannian manifold $(M^{n+1},g)$ with $3\le n+1\le 7$ admits infinitely many closed hypersurfaces with constant mean curvature enclosing half the volume of $M$. This is proposed as a constant-mean-curvature analogue of Yau's conjecture. The source gives no resolution, so the assertion remains open.