Pyber's conjecture on uniformly quasirandom perfect groups of bounded commutator width

Let $n$ be an integer. A group is perfect if it equals its commutator subgroup, and it has commutator width at most $n$ if every element can be written as a product of $n$ commutators. A class of groups is Q.U.P. if every ultraproduct of groups in the class is quasirandom.

Pyber's conjecture. The class of perfect groups with commutator width at most $n$ is Q.U.P.

The conjecture was suggested by László Pyber. It arises because the classes considered in the paper have bounded commutator width; the supplied text gives no resolution of the conjecture.