Kopczyński’s conjecture: closure under union of prefix-independent positional objectives

Determine whether, for all alphabets Σ and all objectives W1, W2 ⊆ Σ^ω that are both prefix-independent and positional (i.e., Eve can play optimally using positional strategies in all games with these winning conditions), the union W1 ∪ W2 is also positional over all games.

Background

The paper revisits a long-standing question due to Kopczyński asking whether the union of two prefix-independent positional objectives remains positional. This closure property fails for non-ω-regular objectives over finite games, but its status over infinite games and for ω-regular languages had remained open historically. The authors later resolve it positively for ω-regular languages when at least one operand is prefix-independent, but the general (non-ω-regular) case remains unresolved.