Is there a model of set theory in which:

Every projectively definable family of sets of reals has an OD or projectively definable member;

Every OD or projectively definable set of reals has the property of Baire.

PS: By a projectively definable family of sets of reals I mean:

There exists a formula $\varphi(\Gamma)$ where $\Gamma$ is supposed to be a set of (tuples) of reals and $\mathcal{F}$ is the family of sets defined by: $$\Gamma \in \mathcal{F} \;{\rm iff} \; \varphi(\Gamma,a)$$ but $\varphi(\Gamma,a) : Q_1 x_1 \dots Q_n x_n Q'_1 z_1 \dots Q'_m z_m \psi(x,z,a,\Gamma)$ with $x$ and $z$ reals and integers respectively, and the $Q_i, Q'_j$ are quantifiers $\in \{ \forall, \exists\}$, $a$ are real parameters. $\psi$ is a $\Delta_0$ formula in the variables $x_i,z_n$ and $\Gamma$.

$\varphi$ and $\psi$ are formulas in the language of arithmetic.

Note that $\Gamma \subset \mathbb{R}^2$ to be precise. An example of $\varphi$ would be: $$\forall x \exists y (x,y) \in \Gamma \wedge [\forall z \forall t (((x,z) \in \Gamma) \wedge ((x,t) \in \Gamma)) \rightarrow z=t]$$ ($\Gamma$ is a function, here $x,y,z,t$ are assumed to be real numbers).

Note that $\mathcal{F}$ may have cardinality $2^{2^{\aleph_0}}$ so we cannot parametrize it by reals.

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