On the finite simple images of free products of finite groups

Given nontrivial finite groups $A$ and $B$, not both of order 2, we prove that every finite simple group of sufficiently large rank is an image of the free product $A\ast B$. To show this, we prove that every finite simple group of sufficiently large rank is generated by a pair of subgroups isomorphic to $A$ and $B$. This proves a conjecture of Tamburini and Wilson.