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

Let G be a group such that $G/Z(G)$ is finite and simple. The non-commuting, non-generating graph $\Xi(G)$ of G has vertex set $G \backslash Z(G)$, 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 $\Xi(G)$ 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.