In "A SPLITTING THEOREM FOR n−REA DEGREES" Shore and Slaman extend the following result of Sacks
If $C$ is r.e., $D \leq_T C$ and $D, C \not\leq_T 0$ then there are sets $C_0, C_1$ such that $C \equiv_T C_0 \oplus C_1$ and $D \not\leq_T C_i$. This result relativizes.
To the case where $C$ is $n$-REA ($n < \omega$). However, the original result of Sacks admits a uniform solution[^1] while the result of Shore and Slaman uses a non-uniform proof. Is it known if Shore and Slaman's generalization to n-REA degrees fails to hold uniformly? (I'm mostly concerned only about the non-relativized case)
[^1]: That is, there is a computable function which, given an r.e. index for $C$ and an index for the reduction computing $D$ from $C$ produces r.e. indexes for $C_0, C_1$. The question regarding the generalization would (obviously) be about indexes as n-REA sets.
