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.

Y Cor has kindly outlined the proof of my 2 conjectures (in comments below) via the Nikolov-Segal Theorem (Annals of Math., 2007). Moreover, Yilong Yang has just found another proof (of my 2nd conjecture) by using his covering properties of finite groups (introduced in his publication in 2016).