Existence of a productive set that is not completely productive

Determine whether there exists a productive set that is not completely productive, where a productive set is a set A ⊆ N admitting a recursive function f such that W_n ⊆ A implies f(n) ∈ A \ W_n, and a completely productive set is a set A ⊆ N admitting a recursive function g such that g(n) ∈ A if and only if g(n) ∉ W_n for every n ∈ N.

Background

The paper defines completely productive sets as a subclass of productive sets characterized by a recursive diagonalization function that determines membership in A through nonmembership in the corresponding recursively enumerable set W_n. Proposition 37 establishes that every completely productive set is productive.

The unresolved issue is whether the converse inclusion is strict: namely, whether productivity alone can occur without complete productivity. Establishing such an example would separate the two notions introduced in the paper; proving that no such example exists would show that the two classes coincide.