Minimal polynomial of the generating tuple of a finite solvable group

Determine whether, for every finite solvable group G generated by g_1,...,g_n, the polynomial q_G associated with the irreducible unitary representations of G is the minimal polynomial of the tuple (g_1,...,g_n).

Background

For a finite group G, the paper defines q_G as the product of the characteristic polynomials associated with the irreducible unitary representations of G. Proposition 5.10 establishes that q_G is the minimal polynomial of the tuple of all nonidentity group elements when the generating set comprises the entire group, and the authors note that analogous reasoning works for broad classes including finite abelian groups and Coxeter groups.

When the specified generators form only a proper subset of G, the individual representation characteristic polynomials need not be irreducible, so the preceding proof does not apply in general. The unresolved question asks whether solvability of G is sufficient to guarantee minimality of q_G for the tuple of the chosen generators.