I conjecture that: Every Finite Homomorphic image of an infinite (with arbitrary cardinality) product of finite solvable groups is solvable -- or at least Not a simple (non-abelian) group.

I can see this conjecture in some cases. but the general case seems very complicated.

Question: Has this problem been investigated ? Thank you.

Edit-- Moreover, I also conjecture that Every Finite Homomorphic image of an INFINITE product G = \prod G_i of finite simple (non-abelian) groups {G_i}, is a product of finite simple groups each of which is isomorphic to some G_i.

It seems that this and little more, was proved very recently by Yilong Yang using his results in his publication in J.Group Theory, 2016. Moreover, it seems that the Nikolov-Segal Theorem (in Annals of Math., 2007) suggested by Y. Cor will yield another proof of my conjecture on finite simple groups. Today, Y Cor has kindly outlined such a proof.