On the Eaton–Moretó conjecture for principal blocks of finite groups

Let G be a finite group and let p be a prime. If P is a nonabelian Sylow p-subgroup of G and $m(P)$ is the smallest non-linear irreducible character degree of P, we prove that there exists χ∈ Irr(G) in the principal p-block of G such that 1<χ(1)ₚ≤ m(P), giving one inequality of the Eaton-Moretó conjecture for principal blocks. This, assuming Dade's Projective conjecture, implies the Eaton-Moretó conjecture for principal blocks.