Let’s call a group $\frac{3}{2}$-generated iff for every $g \in G \setminus \{1\}$ there is some $h \in G \setminus \langle g \rangle$ with $G = \langle g,h \rangle$.
Is every $2$-generated simple group $\frac{3}{2}$-generated?
This happens to be true for finite simple groups and for Tarski monster groups, but is it true in general? I do not know how to prove that, but neither do I know any counterexamples.