Five-point discrete-concavity conjecture under candidate optimizer assumptions

Let $v_s(N)$ be the minimal average standardized Riesz pair-energy on $\mathbb{S}^2$, and let $\ddot v_s(N)=v_s(N-1)-2v_s(N)+v_s(N+1)$. Let $s^\dagger=15.04807\ldots$, and assume that the optimizing configuration for $N=5$ is the regular triangular bipyramid for $s\leq s^\dagger$ and the square pyramid with adjusted height for $s\geq s^\dagger$. Five-point concavity conjecture. Under these assumptions,

$$\forall s\in(-2,\infty):\qquad \ddot v_s(5)<0.$$

The claim is conditional on the stated optimizer assumptions, which are only partly rigorously known, so the resulting evidence is partly rigorous and partly numerical.