The auxiliary Aronszajn-null game characterization conjecture

Let $X$ be a separable Banach space, let $A\subseteq X$ be Borel, and let $\mathbb{A}_{\mathsf{G}}$ be the auxiliary game referred to in the paper, with $\mathbb{A}$ denoting the corresponding class of sets. The notation $\mathbf{I}\not\!\uparrow\!\mathbb{A}_{\mathsf{G}}$ means that player $\mathbf{I}$ has no winning strategy. The auxiliary Aronszajn-null game characterization conjecture.

$$\mathbf{I}\not\!\uparrow\!\mathbb{A}_{\mathsf{G}}$$

if and only if

$$A\in\mathbb{A}.$$

The claim is presented as a reformulation of an earlier conjecture using the equivalence between the two games for player $\mathbf{I}$; its resolution is not given in the supplied text.