In a question and an answer at MO, Joel David Hamkins showed that (if ZFC is consistent) there are models of ZFC in which $V\neq HOD$ and every $\Sigma_2$-definable set has a definable member.

Let $\mathfrak{M}$ be such a model. My question is: Can such a model $\mathfrak{M}$ satisfy further the following:

(*) Every ordinal definable set of reals is Baire (or Lebesgue) measurable?

I suspect very much that the answer is negative, but I would like the confirmation of an expert.