1-to-2 player lift in BC(Σ^0_2) (objectives recognised by deterministic parity automata over infinite alphabets)

Establish whether, for objectives in the class BC(Σ^0_2), positionality over Eve-games implies positionality over all games (i.e., prove the 1-to-2 player lift for BC(Σ^0_2) objectives).

Background

The authors extend their paper beyond ω-regular objectives to topological classes. They highlight BC(Σ0_2) as a natural next step where a lift analogous to their ω-regular result is conjectured but unproved.

References

Conjecture [1-to-2 player lift in $\BCSigma$] Let $W$ be an objective in $\BCSigma$ that is "positional over" "Eve-games". Then, it is "positional" over all games.

— Positional $ω$-regular languages (2401.15384 - Casares et al., 27 Jan 2024) in Conclusions, Positionality of BCΣ languages (Conjecture)