Thank you but the result that I still could not prove is Theorem 46. It is also known that the reciprocal is not valid because in the article Barely Baire Spaces of W. Fleissner and K. Kunen there is a space $X$ such that the Tychonoff power $X^{\omega_{1}}$ is Baire but the box power $X^{\omega_{1}}$ is not Baire.

In fact, later using Cohen's idea (using stationary sets in $\omega_1$), were constructed two Baire metric spaces whose product is not Baire in ZFC. (Applications of stationary sets in topology - William G.Fleissner in Surveys in General Topology 1980, Pages 163-193)

Thanks @Wojowu, I did not know that great result.