The non-commuting, non-generating graph of a finite simple group

Let $G$ be a group such that $G\mathrm{/}Z(G)$ is finite and simple. The non-commuting, non-generating graph $\mathrm{\Xi }(G)$ of $G$ has vertex set $G\setminus Z(G)$, with edges corresponding to pairs of elements that do not commute and do not generate $G$. Complementing our previous investigation of this graph for non-simple groups, we show that $\mathrm{\Xi }(G)$ is connected with diameter at most $5$, with smaller upper bounds for certain families of groups. Using these bounds, we then prove that when $G$ is simple, the diameter of the complement of the generating graph of $G$ has a tight upper bound of $4$, with the exception of at most one group with a graph of diameter $5$.