Analytic sets and Turing determinacy

I wonder whether the following question have a positive answer within $ZFC$.

Question If $\{A_n\}_{n\in \omega}$ is a sequence of analytic sets so that $\bigcup_n A_n=2^{\omega}$, then there must be some $n$ so that $A_n$ has a pointed subset.

A pointed set is a perfect set $P$ of reals in which every member computes the representation of $P$.

Note that $PD$ implies a positive answer of the question.