Generalized pigeonhole principle versus two-randomness

Characterize the implication relationship between GPHP at level Sigma-two and the first-order theory 2-RAN.

Background

The theory 2-RAN is placed between CARD at level Sigma-two and WPHP at level Sigma-two. Its relationship with the generalized pigeonhole principle is unresolved.

References

The first-order theory $\text{2-RAN}$ is between $\text{CARD}(\Sigma_2)$ and $\text{WPHP}(\Sigma_2)$, but how about $\text{GPHP}(\Sigma_2)$ and $\text{2-RAN}$?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in 2025 Section 5