Is there an undetermined Banach-Mazur game in ZF?

logic set-theory general-topology axiom-of-choice infinite-games

This has now been partially answered at Mathoverflow by James Hanson, with the remaining case separately asked; I'm posting this answer to move this question off the unanswered queue (and I've made it CW so I don't get reputation for his work). The answer is affirmative under DC, and currently wide open otherwise.

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