Conjecture on non-proper free boundary min-max hypersurfaces

Let $n$ be the dimension parameter and consider compact smooth domains in $\mathbb{R}^{n+1}$. A free boundary min-max minimal hypersurface is a free boundary minimal hypersurface produced by a one-parameter min-max construction. Non-properness conjecture. There exists a compact smooth domain in $\mathbb{R}^{n+1}$ for which any one-parameter free boundary min-max minimal hypersurface is non-proper. This conjecture asserts that the properness conclusion for the min-max hypersurfaces in the stated theorem is optimal; the paper proposes a geometric domain as a possible example, but does not establish the claim.