Given a (non-computable) $\Pi^0_2$ singleton $Y$ are there Turing incomparable $\Pi^0_2$ singletons $X_0, X_1$ with $Y \equiv_T X_0 \oplus X_1$?

What about the same question for arithmetic reducibility?

Given a (non-computable) $\Pi^0_2$ singleton $Y$ are there Turing incomparable $\Pi^0_2$ singletons $X_0, X_1$ with $Y \equiv_T X_0 \oplus X_1$?

What about the same question for arithmetic reducibility? EDIT: Wait I've solved that one with an arithmeticly minimal singleton.