Turing-invariance of strong relativization for \(\Delta^0_2\)

Determine whether Turing equivalence preserves the property of strongly relativizing \(\Delta^0_2\) at every recursively presented Polish space.

Background

The paper proves that, except for $\Delta^0_1$ and $\Delta^0_2$, strong relativization of the relevant self-dual Kleene pointclasses is preserved under Turing equivalence. It also proves that preservation fails for $\Delta^0_1$. The remaining borderline case $\Delta^0_2$ is explicitly left unresolved.