The maximal 3-term progression conjecture for the 2-dimensional sphere

Let $S^2=\{u\in\mathbb{R}^3:|u|=1\}$ be the 2-dimensional sphere, and for $n\geq 2$ let $\mu_n(S^2)$ denote the maximal number of $3$-term arithmetic progressions in an $n$-element subset of $S^2$. Maximal progression conjecture. For every $n\geq 2$,

$$\mu_n(S^2)=\frac{1}{2}n^2+\begin{cases} 2n-4 & n\bmod 4=0,\\ \frac{5}{2}n-8& n\bmod 4=1,\\ 3n-6 & n\bmod 4=2,\\ \frac{5}{2}n-7 & n\bmod 4=3. \end{cases}$$

The displayed value is attained by suitable subsets of the union of a great circle and the corresponding pair of poles, so the conjecture asserts that the lower bound supplied by those configurations is optimal for every $n$.