Existence of a c.e. set intermediate for Sigma-Feiner and low for Delta-Feiner

Let $A$ be a computably enumerable set. Say that $A$ is intermediate for $\Sigma$-Feiner if it is neither low nor high for $\Sigma$-Feiner**, and say that it is low for $\Delta$-Feiner if it is low for that hierarchy.

C.e. Sigma-intermediate/Delta-low conjecture. There is a computably enumerable set that is intermediate for $\Sigma$-Feiner and low for $\Delta$-Feiner**.

This is a specific instance of differing behavior between the two Feiner hierarchies. The supplied text provides no resolution status beyond its presentation as a conjecture.