The trivial intersection problem for characters of principal indecomposable modules

For a prime p and a finite group G let Φp(G) denote the complex character associated to the projective indecomposable module in characteristic p with trivial head. Let Irr(Φp(G)) denote the set of irreducible characters occurring as constituents in Φp(G). We characterize all finite simple groups which satisfy Irr(Φp(G))∩Irr(Φq(G))={1G} for all primes p≠q.