Question.Is there a Baire subspace $X$ of a Tychonoff power $M^\kappa$ of some separable metrizable space $M$ such that for any countable subset $A\subset \kappa$ the projection $$X_A=\{x{\restriction}A:x\in X\subset M^\kappa\}\subset M^A$$ of $X$ onto $M^A$ in a meager (metrizable separable) space?

Actually, I am interested in the negative answer under the assumption that the Baire space $X$ has countable spread (i.e., does not contain an uncountable discrete subspace).

Let us recall that a topological space $X$ is

$\bullet$ *meager* if $X$ can be written as a countable union of closed subsets with empty interior;

$\bullet$ *Baire* if $X$ contains no non-empty open meager subspace.

**Remark.** I admit that the answer to the Question can depend on some set-theoretic assumptions (like CH or Martin's Axiom).