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 $G = \langle g,h \rangle$.
There is a conjecture by Breuer, Guralnick and Kantor, that a finite group is $\frac{3}{2}$-generated iff all its proper quotients are cyclic.
Is this conjecture known to be true for finite simple groups? A finite simple group has no nontrivial proper quotients, thus according to that conjecture all finite simple groups are $\frac{3}{2}$-generated? Are they all known to be indeed $\frac{3}{2}$-generated, or is that “reduced” version of the conjecture also an open problem?
Yes. That was proved by Robert M. Guralnick and William M. Kantor in "Probabilistic generation of finite simple groups"