The maximum spherical two-distance set conjecture

Let $g(n)$ denote the maximum cardinality of a spherical two-distance set in $\mathbb{R}^n$. Maximum spherical two-distance set conjecture.

$$g(n)=\frac{n(n+1)}{2}\quad\text{if }n\ne(2k+1)^2-3\quad\text{for some }k\in\mathbb{N}.$$

The conjecture reflects the observed pattern that the maximal size is generally $n(n+1)/2$, with the dimensions $(2k+1)^2-3$ as the possible exceptions. Its resolution is not given in the supplied text.