Character degrees of finite p-groups by coclass

Let (Gk|k∈N0)(Gk|k∈N0) be a coclass family of finite p-groups with associated pro-p-group S, and let l be a nonnegative integer. We show that for large k   the number of irreducible characters of GkGk with degree plpl is a polynomial in pkpk of degree less than dim⁡Sdim⁡S; i.e. it has the form fl(pk)fl(pk) for some fl(x)∈Q[x]fl(x)∈Q[x] with deg⁡fl≤dim⁡Sdeg⁡fl≤dim⁡S.