Diagonal degrees for computably enumerable equivalence relations

Characterize the Turing degrees in Diag, determine the reverse-mathematical strength of the assertion that every nontrivial Sigma-zero-one equivalence relation has a diagonal function, and determine the corresponding Weihrauch degree.

Background

A diagonal function avoids the equivalence class of its input for every input. The paper gives partial bounds on the class Diag and poses degree-theoretic, reverse-mathematical, and Weihrauch-theoretic versions of the problem.