Conjectured moduli of the quarter-sphere quadrilateral and its conjugate

Let $Q$ be the quadrilateral on the sphere whose boundary points are

$$z_1=(0,0,1),\quad z_2=(1,0,0),\quad z_3=\Big(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0\Big),\quad z_4=\Big(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0\Big).$$

Let $\widetilde{Q}$ denote its conjugate quadrilateral, and let $M$ denote the modulus of such a quadrilateral.

Conjectured modulus values.

$$M(Q)=\sqrt{2},\qquad M(\widetilde{Q})=\frac{1}{\sqrt{2}}.$$

The exact moduli are not known, but the paper reports considerable computational evidence for these values in the quarter-sphere problem.