Four-dichotomic-qubit compatibility-degree conjecture

Let $s_{\mathbb C}(2,4)$ denote the minimum compatibility degree of four dichotomic measurements on a complex qubit. Four-dichotomic-qubit compatibility-degree conjecture.

$$s_{\mathbb C}(2,4)=\frac{2}{\sqrt{13}}.$$

The value is attained by the displayed computational-basis and equiangular-line measurements, and numerical optimization provides evidence for optimality; a proof remains open.