Is it known if there is a $\Pi^0_2$ singleton of minimal arithmetic degree?
To elaborate a bit, this is asking whether there is a non-arithmetic set $X$ such that for any $Y$ arithmetic in $X$ either $Y$ is arithmetic or $X$ is arithmetic in $Y$ (remember, being arithmetic in another set is equivalent to being computable from some finite jump) with $X$ being the unique element of $2^{\omega}$ satisfying some $\Pi^0_2$ formula.
The $\Pi^0_2$ singletons include all the $\alpha$-REA sets. As I understand it, the question of the existence of an $\omega$-REA set of minimal arithmetic degree is still open and, thus, a natural generalization one might ask about is whether there is a minimal arithmetic degree that's a $\Pi^0_2$ singleton.
I fear I might be missing some way in which the standard construction using (the arithmetic analog of) e-splitting trees might produce one but hopefully someone can tell me if I am.
