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(Σ^0_2) as a natural next step where a lift analogous to their ω-regular result is conjectured but unproved.