Trivial intersection of blocks and nilpotent subgroups

Let p,q be different primes and suppose that the principal p- and the principal q-block of a finite group have only one irreducible complex character in common, namely the trivial one. We conjecture that this condition implies the existence of a nilpotent Hall {p,q}-subgroup and prove that a minimal counter-example must be an almost simple group where pq divides the order of its simple nonabelian normal subgroup. As an immediate consequence we obtain that the conjecture holds true for p-solvable or q-solvable groups. Furthermore, we prove the conjecture in case 2∈{p,q} using the classification theorem of finite simple groups. Finally, we consider the situation that the intersection of an arbitrary p-block with an arbitrary q-block contains only one irreducible character.