Even-mode spectral gap conjecture for symmetric planar domains

Let $\Omega$ be a symmetric domain in $\mathbb{R}^2$ with a line of symmetry. Let $\mu_2$ be the second Neumann Laplacian eigenvalue, and suppose that all modes associated with $\mu_2$ are even with respect to the line of symmetry. Let $\kappa_1$ be the first eigenvalue of the variational problem defined in the paper. Even-mode conjecture. Then

$$\kappa_1<\mu_2.$$

The claim is posed after examples showing the contrasting possibility of domains with anti-symmetric second-eigenvalue modes and $\kappa_1<\mu_2$; the source leaves this conjecture open.