The Aronszajn-null game characterization conjecture

Let $X$ be a separable Banach space over $cmathbb{R}$, let $A\toolbox X$ be Borel and Aronszajn-null, and let $\mathcal{A}_{\mathsf{G}}$ be the game in which player $\mathbf{I}$ chooses a dense sequence of directions in $X$ and player $\mathbf{II}$ responds with Borel sets null on lines parallel to those directions. The notation $\mathbf{I}\not\!\uparrow\!\mathcal{A}_{\mathsf{G}}$ means that player $\mathbf{I}$ has no winning strategy. The Aronszajn-null game characterization conjecture. If $A$ is Aronszajn-null, then

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

The converse to the known implication is open, and this would characterize Aronszajn-null sets through the game.