Uniqueness of finite simple groups by the number of 2-element centralizers

Let $G$ and $H$ be finite simple groups. For a finite group $K$, let $2\text{-}\operatorname{Cent}(K)$ denote the set of centralizers of $2$-element subsets of $K$, and let $|2\text{-}\operatorname{Cent}(K)|$ be its cardinality. The uniqueness conjecture. If

$$|2\text{-}\operatorname{Cent}(G)|=|2\text{-}\operatorname{Cent}(H)|,$$

then $G\cong H$. The conjecture is motivated by calculations with the aid of GAP; its validity for all finite simple groups remains open.