Min-max existence conjecture for embedded constant mean curvature surfaces

Let $M$ be a closed $3$-manifold, and let $S$ be an embedded surface in $M$ with nontrivial homology. Let $\widehat{C}_{[S]}$ be the lower bound defined using the width of the homology class $[S]$. Min-max existence conjecture. There exists $\widehat{H}_{[S]}\geq \widehat{C}_{[S]}\geq 0$ such that, for every $H\in[0,\widehat{H}]$, there exists a smoothly embedded $H$-surface $\Sigma_H$ in the homology class of $S$. The conjecture would generalize the paper's existence theorem because the min-max width satisfies $w([S])\geq g([S])$, and hence $\widehat{C}_{[S]}\geq C_{[S]}$.