Is there a $\Pi^0_2$ singleton that forms a minimal pair with $0''$? That is, is there a set $X$ such that $X$ is the unique solution to $\forall x \exists y \phi(X|_y, x)$, $X$ and $0''$ are incomparable and if $Y \leq_T 0'' \land Y \leq_T X$ then $Y \leq_T 0$.
For some motivation, note that the most common examples of $\Pi^0_2$ singletons are the $\alpha$-REA sets. Because every $\alpha$-REA set is built up in uniform r.e. sets no $\alpha$-REA set forms a minimal pair with $0''$ (induction and use $0''$ at limit stages to convert r.e. indexes to recursive ones).
Even Harrington's construction of a $\Pi^0_2$ singleton not of $\omega$-REA degree builds the $\Pi^0_2$ singleton by stitching together a part that is built computably in $0'$ with parts built computably in $0''$, $0'''$ and so on.
So it's an attractive hypothesis to think that, in some sense, every $\Pi^0_2$ singleton is somehow built up in pieces analagous to the $\omega$-REA situation. I suspect it's false but don't have a proof.
