Existence of a minimal-cover pair with one co-c.e. endpoint among co-Σ0_2 sets

Determine whether there exist co-Σ0_2 sets D and E such that D ≤s E, there is no set Z with D <s Z <s E (so E is a minimal cover of D in the s-degrees), and at least one of D or E is co-c.e. (Π0_1).

Background

The paper proves that there exist co-Σ0_2 sets D and E with D <s E such that no Z satisfies D <s Z <s E, establishing that the Σ0_2 s-degrees (and correspondingly the Σ0_2 Q-degrees) are not dense. The construction yields both endpoints in the co-Σ0_2 class.

The authors note that one cannot require both endpoints to be 2-c.e., since every 2-c.e. set is s-equivalent to a co-c.e. (Π0_1) set and the Π0_1 s-degrees are known to be dense via their correspondence with c.e. Q-degrees. This leaves open whether one can strengthen their result to make at least one endpoint co-c.e. while retaining the minimal-cover property.