A Brauer--Galois height zero conjecture

Recently, Malle and Navarro obtained a Galois strengthening of Brauer's height zero conjecture for principal p-blocks when $p=2$, considering a particular Galois automorphism of order~$2$. In this paper, for any prime p we consider a certain elementary abelian p-subgroup of the Galois group and propose a Galois version of Brauer's height zero conjecture for principal p-blocks. We prove it when $p=2$ and also for arbitrary p when G does not involve certain groups of Lie type of small rank as composition factors. Furthermore, we prove it for almost simple groups and for p-solvable groups.