Generalized Velickovic-Woodin hypothesis

Determine whether the Velickovic-Woodin conclusion holds for every Sigma-one-one(y) set containing an x whose Church-Kleene ordinal exceeds that of y.

Background

This asks whether the theorem’s global supremum hypothesis can be replaced by the existence of one sufficiently complex member relative to the parameter y.

References

Is the following true: For any $\Sigma_11(y)$ set $A$, if there is $x \in A$ such that $\omega_1{CK(x)} > \omega_1{CK(y)}$, then the Velickovic--Woodin theorem holds for $A$?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in 2025 Section 4