Non-analytical reals strongly relativizing \(\Delta^0_1\) at Baire space

Determine whether there exist non-\(\Delta^1_1\) reals that strongly relativize \(\Delta^0_1\) at Baire space \(N\), and whether continuum many such reals exist.

Background

The paper proves that every $\Pi^0_1$-singleton in Baire space strongly relativizes $\Delta^0_1$ at $N$. It also notes that every $\Delta^1_1$ real is computed by a $\Pi^0_1$-singleton, while not every $\Delta^1_1$ real is itself a $\Pi^0_1$-singleton. This motivates asking whether the class of strongly relativizing reals at $N$ extends beyond the $\Delta^1_1$ reals and how large that extension might be.