The quarter-sphere mixed-boundary spectral conjecture

Let $Q\subset{\Bbb S}^2$ be the quarter-sphere

$$Q=\{(\phi,\theta):0<\phi<\pi/2,\ 0<\theta<\pi\}.$$

Split its boundary as $\partial Q=\overline{\partial_1Q\sqcup\partial_2Q}$, where

$$\partial_1 Q=\{(0,\theta):|\theta-\pi/2|<\pi/4\}\cup\{(\pi/2,\theta):0<\theta<\pi/2\},$$ $$\partial_2 Q=\{(0,\theta):|\theta-\pi/2|>\pi/4\}\cup\{(\pi/2,\theta):\pi/2<\theta<\pi\}.$$

Let $\Lambda_1$ be the first eigenvalue of the mixed problem

$$-\Delta u=\Lambda u\quad\text{on }Q,\qquad u|_{\partial_1Q}=0,\qquad (\partial u/\partial n)|_{\partial_2Q}=0.$$

Quarter-sphere spectral conjecture. One has

$$\Lambda_1\geq 2.$$

This inequality is the numerical spectral reduction proposed for the unresolved Bolza-surface eigenvalue conjecture; proving it would establish the preceding equality.