The Baire power conjecture for Black's winning strategy

Let $X$ be a topological space, let $\pi(X)$ be the least cardinality of a $\pi$-base for $X$, and let $\operatorname{BM}(X)$ denote the Banach–Mazur game on $X$. For positive cardinals $\lambda$ and $\kappa$, let

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

mean that whenever $\pi(X)\leq\lambda$ and the box-product power $X^{\kappa}$ is Baire, Black has a winning strategy in $\operatorname{BM}(X)$. Baire power conjecture. For each $\lambda\geq 1$ there is a $\kappa\geq 1$ such that

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

is true. This conjecture asks whether, for every bound on the $\pi$-weight, some fixed box power being Baire guarantees Black's determinacy in the original game; the supplied text gives no resolution.