Five-point minimizer phase-transition conjecture for the frame potential

Let $\operatorname{FP}_{p,5,2}(X)$ be the $p$th frame potential of a five-point configuration $X$ in $\mathbb{R}^2$, and let $X_5^\perp$, $X_5^{(h)}$, $Y(\alpha)$, and $Z(\alpha)$ be the configurations defined in the paper. Let $\alpha(p)$ denote the parameter selected by minimization, and set $p_1=1.77766251887019$ and $p_2=1.78329970946521$. Five-point minimizer conjecture. The absolute minimizer is

$$\begin{cases} X_5^\perp & 0\le p\le p_1,\\ Y(\alpha(p)) & p_1\le p\le p_2,\\ Z(\alpha(p)) & p_2\le p\le 2,\\ X_5^{(h)} & p\ge 2. \end{cases}$$

Moreover, this minimizer is unique up to rotation and antipodal reflections for any $p$ not at an endpoint of the intervals. The conjecture gives a detailed description of the numerically observed phase transitions for five points; the stated transition values are given to precision $10^{-14}$, and no proof is supplied in the source.