Kechris's universality conjecture for Turing equivalence

A countable Borel equivalence relation is an equivalence relation on a Borel subset of $2^\omega$ whose equivalence classes are countable and whose relation is Borel. A Borel reduction from $(X,\sim_X)$ to $(Y,\sim_Y)$ is a Borel function $f\colon X\to Y$ such that $x\sim_X y$ if and only if $f(x)\sim_Y f(y)$. An equivalence relation is universal if every countable Borel equivalence relation is Borel reducible to it. Kechris's conjecture. Turing equivalence is a universal countable Borel equivalence relation. The paper explains that this conjecture is false for the stronger setting of locally countable Borel quasi-orders, while the status of the conjecture for countable Borel equivalence relations itself is not resolved here.