Dixon's strong 2-generation conjecture for finite simple groups

Let $(G_n)$ be any sequence of finite simple groups such that $|G_n|$ tends to infinity with $n$. Let $\mathbb{P}_2(G)$ denote the probability that two randomly chosen elements generate $G$. Dixon's conjecture. The probability satisfies

$$\mathbb{P}_2(G_n) \longrightarrow 1 \quad\text{as } n \longrightarrow \infty.$$

This extends Netto's asymptotic generation prediction from alternating groups to arbitrary sequences of finite simple groups whose orders tend to infinity.