Kopczyński's finite-union conjecture for half-positional winning conditions

Let a winning condition be a set of infinite plays in an arena. It is half-positional if, for every arena, either Alice has a positional winning strategy or Bob has a winning strategy, not necessarily positional; it is prefix-independent if membership is unchanged after removing or adding finite prefixes.

Kopczyński's conjecture. Prefix-independent half-positional winning conditions are closed under finite unions.

The conjecture is refuted for finite arenas by the paper, which constructs two prefix-independent bi-positional winning conditions whose union is not half-positional. It remains open for infinite arenas.