Restrictions of characters in p-solvable groups

Let G be a p-solvable group, P≤G a p-subgroup and χ∈Irr(G) such that χ(1) _p≥|G:P| _p. We prove that the restriction χ _P is a sum of characters induced from subgroups Q≤P such that χ(1) _p=|G:Q| _p. This generalizes previous results by Giannelli–Navarro and Giannelli–Sambale on the number of linear constituents of χ _P. Although this statement does not hold for arbitrary groups, we conjecture a weaker version which can be seen as an extension of Brauer–Nesbitt's theorem on characters of p-defect zero. It also extends a conjecture of Wilde.