The multiboard Gale–Stewart determinacy conjecture

Let $\lambda\geq 2$ be a cardinal and let $A\subseteq{}^{\omega}\lambda$. In the multiboard Gale–Stewart game

$${\sf GS}_{\lambda}(A,\kappa),$$

the players play the ordinary Gale–Stewart game on $\kappa$ boards simultaneously; ONE wins if at least one resulting board is a win for ONE, while TWO wins if TWO wins on every board. For cardinals $\lambda$ and $\kappa$, let

$${\sf D}(\lambda,\kappa):$$

mean that for every $A\subseteq{}^{\omega}\lambda$, the game

$${\sf GS}_{\lambda}(A,\kappa)$$

is determined. Multiboard Gale–Stewart determinacy conjecture. For each cardinal $\lambda\geq 2$ there exists a cardinal $\kappa\geq 1$ such that

$${\sf D}(\lambda,\kappa)$$

holds. The conjecture rules out the possibility that, for some alphabet size, every finite or transfinite number of boards admits an undetermined multiboard game; the supplied text gives no resolution.