In Feng, Magidor, and Woodin "Universally Baire Sets of Reals", they show that if $A$ is a $\mathbf{\Pi}_2^1$ set and $U$ and $V$ are any pair of trees witnessing the universal baireness of $A$, then in all generic extension, one continues to have $A = p[U]$, where $A$ as defined by the $\mathbf{\Pi}_2^1$.

Is this true for $\mathbf{\Pi_3^1}$ (or more generally any higher projective sets)? Rather than any pair of trees as in the $\mathbf{\Pi}_2^1$ case, if one assumes all $\mathbf{\Pi}_3^1$ sets are universally baire, does there exists some pair of trees $T$ and $U$ such that $T$ continues to represent a given $\mathbf{\Pi}_3^1$ set $A$ in all generic extensions? What about, if one fix a particular forcing $\mathbb{P}$, can one find pairs of trees $T$ and $U$ that continue to represent a particular $\mathbb{\Pi}_3^1$ set in $\mathbb{P}$ extensions.

To be a bit more precise. Suppose $\varphi$ is a $\Pi_3^1$ formula possible using some reals as parameters. With appropriate large cardinals, all $\mathbf{\Pi}_3^1$ sets are universally Baire. Hence there are trees $T$ and $U$ such that

$V \models (\forall x)(\varphi(x) \Leftrightarrow x \in p[T])$ and $V \models (\forall x)(\neg \varphi(x) \Leftrightarrow x \in p[U])$

If $\mathbb{Q}$ is a forcing, then $1_\mathbb{Q} \Vdash_{\mathbb{Q}} ...$ the same two statements.

The question is for any $\mathbf{\Pi}_3^1$ formula $\varphi$ can one find trees $T$ and $U$ witnessing universally Baireness with the additional property that:

For all forcings $\mathbb{Q}$, $1_\mathbb{Q} \Vdash (\forall x)(\varphi(x) \Leftrightarrow x \in p[T])$.

This is what I mean by a $\mathbf{\Pi}_3^1$ sets being represented by the same tree in all generic extensions.

Thanks for any information that can be provided.