We show that there exist downwards properly Σ⁰₂ (in fact noncuppable) e-degrees that are not high. We also show that every high e-degree bounds a noncuppable e-degree.
"Properly Σ⁰₂ enumeration degrees and the high/low hierarchy." J. Symbolic Logic 71 (4) 1125 - 1144, December 2006. https://doi.org/10.2178/jsl/1164060448