Fejes Tóth's conjecture on the sum of acute angles

Let $x_1, x_2,\dots, x_N$ be (not necessarily distinct) points on the sphere $\mathbb S^{d}$, and let $\theta(x_i,x_j)$ denote their non-obtuse geodesic angle. The discrete energy is

$$\sum_{i,j=1}^N \theta(x_i,x_j).$$

Fejes Tóth's conjecture. This energy is maximal when $x_1,\dots,x_{d+1}$ are mutually orthogonal and $x_k=x_{k-(d+1)}$ for every $k$ with $d+1<k\le N$. Equivalently, periodically repeated elements of an orthonormal basis maximize the sum of non-obtuse angles. The conjecture was stated by Fejes Tóth on $\mathbb S^2$ and generalized to all $d\ge 1$; it is proved for $d=1$ and remains open for $d>1$, including the case $N=k(d+1)$.