Multiplicity-one conjecture for smooth fixed-topology min-max surfaces

Let $g$ be a Riemannian metric and let a min-max sequence from a sweepout of genus $g$ surfaces converge as a varifold to

$$\sum_{i=1}^k n_i\Gamma_i,$$

where the pairwise disjoint, smooth, closed, embedded minimal surfaces $\Gamma_i$ have positive integer multiplicities $n_i$. A minimal surface is two-sided when it has a globally defined normal direction. Multiplicity-one conjecture. In the setting of the multi-parameter min-max theorem, if the metric $g$ is bumpy, then $n_i=1$ for every two-sided $\Gamma_i$. More generally, for any metric $g$, every two-sided $\Gamma_i$ occurring with multiplicity $n_i>1$ is stable and has a non-trivial Jacobi field. Multiplicity one is known in related Almgren–Pitts settings through regularized area methods, but remains open here for smooth sweepouts of fixed topological type.