Optimistic uniform universality conjecture for universal countable Borel equivalence relations

Let $E$ be a countable Borel equivalence relation. It is universal if every countable Borel equivalence relation Borel reduces to $E$. Given a generating family of Borel functions, $E$ is uniformly universal with respect to that family if every countable Borel equivalence relation has a uniform Borel reduction to $E$ relative to the generating families.

Optimistic uniform universality conjecture. If $E$ is a universal countable Borel equivalence relation, then $E$ is uniformly universal with respect to every way of generating $E$.

The paper calls this conjecture “ridiculously optimistic.” It is motivated by the fact that every known universal countable Borel equivalence relation has been established as universal by a proof uniform in this sense.