Concavity-defect monotonicity conjecture for Riesz energies

Let $v_s(N)$ be the minimal average standardized Riesz pair-energy on $\mathbb{S}^2$, and let ${\cal C}_+(s)$ be the set of integers $N$ at which strict local concavity fails, so that $\ddot v_s(N)\geq0$. Concavity-defect monotonicity conjecture. For $s\in\{0,1,2,3\}$, the map $N\mapsto v_s(N)$ is not locally strictly concave, and the sets of $N$-values where strict concavity fails expand set-theoretically as $s$ increases. The assertion is empirical and concerns the organization of observed concavity defects; its general validity remains open.