Symmetric-group quotients forcing symmetric subgroups

Prove or disprove that, for every positive integer n, there exists a positive integer N such that every finite group with a quotient isomorphic to S_N contains a subgroup isomorphic to S_n.

Background

This conjecture is presented as a proposed counterexample to the possibility that quotient-to-subgroup implications never force a non-abelian subgroup. It asserts that sufficiently large symmetric quotients universally force smaller symmetric subgroups in finite extensions.

The authors state that the conjecture is equivalent to requiring, for every finite group H_2, the existence of some finite group H_1 satisfying H_1⇒H_2.