Equality of the transition heights for symmetric and antisymmetric second Neumann eigenfunctions

Let $h_0$ and $h_1$ be the positive heights from part (2) of Theorem, where the ordering of the symmetric and antisymmetric second Neumann eigenvalues changes as the height varies. Transition-height equality conjecture. The constants $h_0$ and $h_1$ in part (2) of Theorem are equal. The equality would identify a single transition height separating the regimes in which the second Neumann eigenfunctions are symmetric or antisymmetric about the $x$-axis. The supplied text does not indicate whether this conjecture has been resolved.