We prove that for every Σ02 enumeration degree b there exists a noncuppable Σ02 degree a > 0e such that b' ≤c a' and a'' ≤c b''. This allows us to deduce, from results on the high/low jump hierarchy in the local Turing degrees and the jump preserving properties of the standard embedding ι : 𝒟T → 𝒟e, that there exist Σ02 noncuppable enumeration degrees at every possible—i.e., above low1—level of the high/low jump hierarchy in the context of 𝒟e.
Charles M. Harris. "On the jump classes of noncuppable enumeration degrees." J. Symbolic Logic 76 (1) 177 - 197, March 2011. https://doi.org/10.2178/jsl/1294170994