Suppose $T$ is a tree on $\omega \times \omega \times \delta$ for some ordinal $\delta$ is a homogeneous tree (with some coherent set of measures witnessing homoegeneity). ($T$ can have additional properties if it is helpful for getting the tree $S$ with the property I want below.)

Under the appropriate large cardinal assumptions, the set

$(\forall y \in {}^{\omega}\omega)p[T] = \{x \in {}^\omega\omega : (\forall y)(\exists f)((x,y,f) \in [T])\}$

is also homogeneous Suslin. (I believe this result is essentially part of Martin-Steel proof of projective determinacy.) So there is some homogeneous tree $S$ so that $p[S] = (\forall y)p[T]$.

My question is that if $\mathbb{P}$ is a very small forcing (say size $2^{\aleph_0}$), can one choose $S$ so that $1_{\mathbb{P}} \Vdash p[S] = (\forall y)p[T]$?

I think this result is true if $T$ is a homogeneous tree representing a projective set (See my earlier question Universally Baire Tree Representation of Projective Sets). I have not seen a proof of generic absoluteness for projective sets, but I think this may be one way of getting projective absoluteness for forcings of certain sizes. My question is essentially whether some type of generic absoluteness holds for formulas of the form $(\forall y)A(x)$ where $A$ is a homogeneous Suslin set.

Thanks for any information or references about this question.