It is known that the product of two Baire spaces can be meager.
On the other hand, by a recent result of Li and Zsilinszky the product of two Baire spaces is Baire if one of the spaces is countably cellular (i.e., does not contains an uncountable family of pairwIse disjoint open sets).
This suggests the following
Problem. Let $H$ be a closed normal subgroup of a topological group $G$, and $G/H$ be the quotient topological group. Assume that the spaces $H$ and $G/H$ are Baire and one of them is countably cellular. Is the space $G$ Baire?
It seems that this question is open even for locally convex linear topological spaces.