Principally separated non-separated solvable groups

If is a set of primes, a finite group is called block -separated if for every two distinct irreducible complex characters α, ∈ Irr( ) there is a prime ∈ such that α and are in different -blocks. The group is called principally -separated if the above holds whenever = 1 . Bessenrodt and Zhang conjectured that if is a solvable principally -separated group then is -separated. We construct a family of counter-examples to this conjecture.