Uniform universality characterization for countable Borel equivalence relations

A countable Borel equivalence relation is an equivalence relation on a standard Borel space whose equivalence classes are countable. A relation $E$ is universal if every countable Borel equivalence relation Borel reduces to $E$. Given a generating sequence of Borel functions, $E$ is uniformly universal with respect to that sequence if every countable Borel equivalence relation, presented by its own generating sequence, admits a uniform Borel reduction to $E$.

Uniform universality characterization. A countable Borel equivalence relation is universal if and only if it is uniformly universal with respect to every way it can be generated.

Uniform universality is intended to capture the uniform methods used in known proofs of universality, but the equivalence between universality and uniform universality for every generating presentation remains open.