theory – Breuer-Guralnick-Kantor conjecture and infinite generated groups in 3/2

A group $ G $ is called $ frac {3} {2} $-generated if each non-trivial element is contained in a generating pair, that is to say rangle = G. $$

See this beautiful Scott Harper poster.
Proposal: Yes $ G $ is $ frac {3} {2} $-generated then each own quotient of $ G $ is cyclical (proof).
Conjecture (B.G.K.): A finite group is $ frac {3} {2} $-if generated if each correct quotient is cyclical.
Theorem (G.K.): Each simple finite group is $ frac {3} {2} $-généré.

Question: Can the conjecture above be extended to finite groups?
In other words: is there a counterexample (known) for such groups?

Of course, any simple group that is not finished (like the alternating infinite group) $ A _ { infty} $) has all appropriate cyclic quotients, but is not $ frac {3} {2} $-généré.