As Taras Banakh noted, there is a theorem for Tychonoff powers,

Theorem Let $\kappa\geq \omega$. If $X^{\omega}$ is Baire, then $X^{\kappa}$ is Baire, where the powers are considered in the Tychonoff product.

Now, if we assume Scheepers' theorem we have the following

Corollary Let $X$ be a topological space, if for all cardinal $\kappa$, the box power $X^{\kappa}$ is a Baire space then the Tychonoff power $X^{\omega}$ is a Baire space.

As Taras Banakh noted, there is a theorem for Tychonoff powers,

Theorem Let $\kappa\geq \omega$. If $X^{\omega}$ is Baire, then $X^{\kappa}$ is Baire, where the powers are considered in the Tychonoff product.

Now, if we assume Scheepers' theorem we have the following

Corollary Let $X$ be a topological space, if for all cardinal $\kappa$, the box power $X^{\kappa}$ is a Baire space then the Tychonoff power $X^{\omega}$ is a Baire space.

I think this can help solve the next question

Can one prove in ZFC that if a box product of a collection of Baire spaces is Baire, then its Tychonoff product is Baire?