Separation between strong relativization at Baire space and at Cantor space

Determine whether there exist reals that strongly relativize \(\Delta^0_1\) at Baire space \(N\) but fail to strongly relativize \(\Delta^0_1\) at Cantor space \(2^N\).

Background

The paper constructs a real $f\in N$ that strongly relativizes $\Delta^0_1$ at $N$ while a Turing-equivalent real $A\in 2^N$ does not strongly relativize $\Delta^0_1$ at $N$, and indeed not even at $N$. The subsequent question asks whether a different kind of separation—between the two parameter or target spaces themselves—can occur for the same real.