Generic properness conjecture for free boundary min-max hypersurfaces

Let $\Omega$ be a compact domain in $\mathbb{R}^{n+1}$ with analytic boundary $\partial\Omega$. A free boundary min-max minimal hypersurface is a free boundary minimal hypersurface produced by a min-max construction. Generic properness conjecture. For a generic compact domain $\Omega\subset\mathbb{R}^{n+1}$ with analytic boundary $\partial\Omega$, any free boundary min-max minimal hypersurface must be proper. The conjecture proposes that although non-proper min-max hypersurfaces may occur for specially chosen domains, properness should hold for generic analytic domains; no resolution is given in the paper.