Pi-one class with maximally almost disjoint recursive members

Construct a Pi-zero-one class with infinitely many recursive members such that its recursive members are maximally almost disjoint among recursive sets.

Background

The question combines effective closedness with a maximal almost-disjointness requirement on the recursive members of the class.

References

Is there a $\Pi0_1$ class $C$ such that $|C\cap {\rm Rec}|=\infty$ and $C\cap {\rm Rec}$ is maximally almost disjoint among the recursive sets?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in Section 2, subsection Computability Theory