Studying the relation between Measure Theory and Baire Category in the real line I came across the following doubt. It is well known that if a subset $A\subseteq\mathbb{R}$ satisfies that every $B\subseteq A$ is Lebesgue measurable, then $A$ must have measure zero, see for example this answer or this other one.
My question is: is the same true with respect to the Baire category ideal? That is, assume that $A\subseteq\mathbb{R}$ satisfies that every $B\subseteq A$ has the Baire property. Can we conclude that $A$ is meagre in $\mathbb{R}$?
I know that, using the same Vitali sets of the answers mentioned before, one can show that there are subsets of $\mathbb{R}$ without the Baire property (see for example here), but I'm looking for something slightly stronger. I tried to adapt the previous proofs to the Baire category setting but could not get anywhere, so any help is much appreciated.
