Equivalence of unravelability and universal unravelability

Determine whether unravelability implies universal unravelability: prove or refute that for every game tree T on a set A and every payoff set P ⊆ [T], if (T, P) is unravelable, then for every covering (π, ·): T′ → T the lifted game (T′, π−1(P)) is unravelable.

Background

Universal unravelability is a strengthening of unravelability designed to yield a σ-algebra of payoff sets and to support the inverse limit construction used in the countable-union step of the proof of Borel determinacy.

While the paper proves that universally unravelable payoff sets form a σ-algebra and that closed games are universally unravelable, it remains unclear whether unravelability itself already implies universal unravelability. Resolving this would clarify whether the stronger notion is strictly stronger or equivalent to Martin’s original unravelability.