The class of groups all of whose subgroups with lesser number of generators are free is generic

It is shown that, in a certain statistical sense, in almost every group with $m$ generators and $n$ relations (with $m$ and $n$ chosen), any subgroup generated by less than $m$ elements (which need not belong to the system of generators of the whole group) is free. In particular, this solves Problem 11.75 from the Kourov Notebook. In the proof we introduce a new assumption on the defining relations stated in terms of finite marked groups.