Conjecture on generalized energy-distance bounds for bivariate spherical copulas

Let $\beta\in(0,2)$, let $\mathcal{C}$ be the set of all bivariate copulas, and for $C\in\mathcal{C}$ let $\boldsymbol X$ and $\boldsymbol Y$ be independent samples from $C$. Define

$$S_\beta(C,C)={\mathbb E}\left(\|\boldsymbol X-\boldsymbol Y\|_2^\beta\right).$$

The conjecture concerns the optimization problem

$$\max_{C\in\mathcal{C}}S_\beta(C,C).$$

Generalized energy-distance copula conjecture. The solution is attained by the circular bivariate $L_p$-norm spherical copula with $p=3-\beta$ for arbitrary $\beta\in(0,2)$. For $\beta=1$, the bivariate spherical copula is known to solve this optimization problem, while the general case $\beta\ne1$ remains open and is supported by numerical experiments.