Finite-support minimizer conjecture for causal variational principles on the sphere

Let $S^2$ be the sphere, let $\tau\geq 1$ be the parameter of the variational problem, and let $\mathfrak{M}$ denote the class of admissible Borel measures on the sphere. A weighted counting measure is a measure of the form

$$\int_{S^2} f\,d\rho:=\sum_{i=1}^{m}\rho_i f(x_i)$$

for points $x_1,\ldots,x_m\in S^2$ and weights $\rho_i\geq 0$ with $\sum_{i=1}^{m}\rho_i=1$.

Finite-support minimizer conjecture. For any $\tau\geq 1$, there is a minimizer $\rho\in\mathfrak{M}$ of the variational problem on the sphere which is a weighted counting measure supported at $m_0$ points.

Numerical minimization suggests that the minimum becomes constant once the number of summands reaches $m_0$, so that a minimizer with finite weighted support should also minimize over all admissible Borel measures. The conjecture concerns the existence of such a finitely supported minimizer for every $\tau\geq 1$.