Breuer–Guralnick–Kantor spread-one conjecture for finite groups

Let $G$ be a finite group. A proper quotient means a quotient $G/N$ with $1\neq N\trianglelefteqslant G$, and $s(G)$ denotes the largest integer $k$ such that any $k$ nontrivial elements of $G$ are simultaneously contained in generating pairs. Breuer–Guralnick–Kantor conjecture. Then

$$s(G)\geqslant 1 \quad\Longleftrightarrow\quad \text{every proper quotient of $G$ is cyclic}.$$

The condition is necessary in general, and the source states that the conjecture was settled by Burness, Guralnick and Harper in 2021.