The explicit leading constant for optimal spherical $\mathbb{L}_2$-discrepancy on $\mathbb{S}^2$

Let $\mathbb{S}^2$ be the unit sphere, let $\operatorname{D_C}^{\mathbb{L}_2}(\mathbb{S}^2;N)$ denote the optimal spherical-cap $\mathbb{L}_2$-discrepancy, and let $A_2$ be the constant

$$A_2=\sqrt{\frac{3}{2}\left(\frac{8\pi}{\sqrt{3}}\right)^{1/2}\left[-\operatorname{\zeta}(-1/2)\right]\operatorname{L}_{-3}(-1/2)}=0.44679728350408\dots.$$

The spherical discrepancy constant conjecture. If both the fundamental optimal Riesz-energy conjecture and the Kuijlaars--Saff conjecture hold, then

$$\operatorname{D_C}^{\mathbb{L}_2}(\mathbb{S}^2;N)\sim A_2N^{-1/2-1/(2d)}+\cdots\qquad\text{as }N\to\infty.$$

This gives an explicit numerical leading constant in dimension two, conditional on the two energy conjectures.