Recursion-theoretic proof of the Velickovic-Woodin theorem

Develop a recursion-theoretic proof of the Velickovic-Woodin theorem concerning hyperdegrees of elements of a Sigma-one-one set.

Background

The theorem gives a four-real hyperdegree bound under an unboundedness condition on admissible ordinals. The problem asks whether its proof can be carried out using recursion-theoretic methods.