Reducing towers of principal fibrations

Consider a tower of principal fibrations That is, Ei+1 is the pullback of Ei→Ri and the path fibration PRi → Ri . The question arises as to whether or not the tower can be shortened, that is, whether or not En+1→ B is fiber homotopically equivalent to a nice fibration E → B. If “nice” is taken to mean “principal” then sufficient conditions are known. They involve connectivity assumptions on the Ei . In this paper “nice” is taken to mean “D-relatively principal” for some space D.