Some conjectures on Brauer character degrees

Let G be a finite group and let p be a prime. Willems conjectured that the sum of the squares of all Brauer character degrees of a finite group G is bounded below by the p^′-part of the order of G. In this note, we propose a ‘projective’ version of this conjecture and prove it for p-solvable groups. Using this, we manage to reduce the proof of Willems conjecture to a question concerning quasi-simple groups. We also introduce a projective version of a conjecture due to Martínez-Pérez and Willems on projective indecomposable characters which, if true, implies the veracity of the projective version of the Willems conjecture above.