Is there a Borel function $f:2^\omega\to\omega^\omega$ such that for every nonempty closed perfect set $P\subseteq 2^\omega$, $f|P$ is a dominating family of functions in $\omega^\omega$?

This is a refinement of the question I asked in: A Borel perfectly everywhere surjective function on the Cantor set. I think the argument which yields a "no" answer there would also give that there is no function $2^\omega\to\omega^\omega$ with comeager, or possibly even nonmeager, image on every perfect set. But while every nonmeager family of functions in $\omega^\omega$ is unbounded, the converse (even for dominating families) is not true.