Generic polynomials are descent-generic

Let g(X) is an element of K(t(l),...,t(m))[X] be a generic polynomial for a group G in the sense that every Galois extension N/L of infinite fields with group G and K less than or equal to L is given by a specialization of g(X). We prove that then also every Galois extension whose group is a subgroup of G is given in this way.