Non-determinacy of the cut-and-choose game on a Bernstein subset of the double arrow space

Let $B\subset [0,1]$ be a Bernstein set, and let $\mathbb{A}$ denote the double arrow space. Define

$$Y=\{(x,1)\in \mathbb{A}:x\in B\}.$$

Non-determinacy problem. The game $G_{\omega}(Y,\mathbb{A})$ is not determined.

This asks whether neither player has a winning strategy in the specified cut-and-choose game; the supplied context does not state whether the problem is solved or remains open.