Fix a forcing notion $\mathbb{P}$. Say that a formula $\varphi(x)$ with parameters is *$\mathbb{P}$-enforceable* if there is some countable set $\mathcal{D}$ of dense sets in $\mathbb{P}$ such that for every $\mathcal{D}$-generic filter $G\subseteq\mathbb{P}$ there is some $p\in G$ such that for every $\mathcal{D}$-generic filter $H\subseteq \mathbb{P}$ with $p\in H$ we have $$\varphi(G)\iff\varphi(H).$$ Note that this all takes place within $V$ itself, and since $\mathcal{D}$ is required to be countable enforceability isn't trivial.

In general, we don't always have that $\varphi$ is $\mathbb{P}$-enforceable$^*$ - however, for reasonably natural $\varphi$ and $\mathbb{P}$ it does tend to be the case that $\varphi$ is $\mathbb{P}$-enforceable. I'm interested in the particular case when $\mathbb{P}$ is Cohen forcing $\operatorname{Fin}(\omega, 2)$ and $\varphi$ is projective with only real parameters. I've heard it stated repeatedly that the usual suspects guarantee tameness, but I've never seen a citation for this and the proof isn't immediate to me. So my question is:

Is it actually the case that, assuming reasonable large cardinals, every projective formula $\varphi$ with only real parameters is $\mathbb{P}$-enforceable? If so, what's a citation for this fact?

$^*$Here's one counterexample - indeed, where $\varphi$ is parameter-free and projective. Assume $V=L$, take $\mathbb{P}$ to be the usual Cohen forcing, let $a$ be a projective bijection from $\mathbb{R}$ to $\omega_1$, and let $b$ be a bijection between the set of countable sets of dense subsets of $\mathbb{P}$ and $\omega_1$. We can define by recursion a projective injection $f:\alpha\rightarrow\mathbb{R}$ such that $f(\alpha)$ is $b^{-1}(\beta)$-generic for all $\beta\le\alpha$. Now let $\varphi(x)$ be the projective formula "$x\in \operatorname{ran}(f)$ and $f^{-1}(x)$ is a limit ordinal." Of course, this sort of nonsense relies on wild projective sets, so large cardinals rule it out.