Iterations of quasi-uniform hyperspace constructions

Given a quasi-uniform space (X,U), we study its Hausdorff quasi-uniformity U-H on the set P-0(X) of nonempty subsets of the set X. In particular we are concerned with the question whether at a certain finite stage iterations of the described Hausdorff hyperspace construction applied to two distinct quasi-uniformities on X will necessarily lead to hyperspaces carrying distinct induced topologies.