A strengthened Gerzon bound for spherical even-distance sets

Let $n\geq 1$ be a positive integer and let $s>0$ be an even integer. Let $\mathcal S\subseteq \mathbb S^{n-1}$ denote a spherical $s$-distance set with inner products $t_1,\ldots,t_s$ satisfying

$$t_1+\ldots+t_s\geq 0.$$

Strengthened Gerzon bound. Then

$$|\mathcal S|\leq {n+s-1\choose s}.$$

This is presented as a natural strengthening of the paper's preceding theorem. The claim concerns the maximum size of spherical distance sets under the additional nonnegative-sum condition on their inner products; no resolution is supplied in the given text.