Restrictions of characters in p-solvable groups

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