Twin-volume conjecture for constant mean curvature hypersurfaces

Let $(M^{n+1},g)$ be a closed Riemannian manifold with $3\leq n+1\leq 7$, and let $v\in(0,\operatorname{Vol}(M))$. Twin-volume conjecture. The manifold $M$ admits at least two distinct closed constant mean curvature hypersurfaces enclosing volume $v$. This is stated as a counterpart of the Twin Bubble Conjecture with a volume constraint replacing prescribed mean curvature; its status is not resolved in the supplied source.