The quarter-sphere first-eigenvalue inequality conjecture

Let $Q$ be either of the two isospectral mixed Dirichlet-Neumann problems on a quarter-sphere shown in the source, and let $Q_{a,2}$ be the axisymmetric problem with the Dirichlet condition on one half of the large semicircle and the Neumann condition on the other half. Denote their first Laplace eigenvalues by $\lambda_1(Q)$ and $\lambda_1(Q_{a,2})$. The quarter-sphere eigenvalue inequality conjecture.

$$\lambda_1(Q)>\lambda_1(Q_{a,2}).$$

The inequality is needed to complete the proof of sharpness of the genus-2 eigenvalue bound in the authors' related work; the source presents it as an assertion still requiring verification and gives no resolution.