Pointed-perfect-set conjecture for increasing Borel approximations to Turing equivalence

Let $(E_n)_{n\in\omega}$ be Borel equivalence relations on $2^{\omega}$ such that $E_n\subseteq E_{n+1}$ for every $n$, and suppose $\equiv_T=\bigcup_{n\in\omega}E_n$. A pointed perfect set is a perfect set with the coding property used in the paper. Pointed-perfect-set conjecture. There is a pointed perfect set $P$ and an index $i$ such that

$$E_i\mathbin{\restriction}P=\equiv_T\mathbin{\restriction}P.$$

The conjecture would give a negative answer to the paper's question about whether Turing equivalence can be written as a nontrivial increasing union of Borel equivalence relations. No resolution is supplied.