Dispersive degrees versus AEWG degrees

Determine whether every dispersive set is almost-everywhere weakly generic (AEWG), equivalently whether there exists a dispersive set that is not AEWG, including within the recursively enumerable degrees.

Background

A set is AEWG if almost every oracle computes a weakly 1-B-generic set relative to it. The paper notes that AEWG implies dispersiveness, a property defined using the Hausdorff metric on coarse similarity classes. The unresolved issue is whether the converse holds. The paper explicitly emphasizes that this remains unknown even among recursively enumerable degrees and presents the question as a relativized degree-theoretic problem.

References

But it is still unknown whether being dispersive implies being AEWG even within the r.e. degrees.

— Almost-everywhere computation of weak generics relative to r.e. sets  (2609.17994 - zhao, 16 Sep 2026) in Section 1, immediately after the definition of AEWG; Question 1

It is still unknown whether being dispersive implies being AEWG for an r.e. degree. So the following question is the next step towards Question \ref{q1}: If $A$ is an r.e. set which is not a.e.d., must $A$ be AEWG?

— Almost-everywhere computation of weak generics relative to r.e. sets  (2609.17994 - zhao, 16 Sep 2026) in Section 1, paragraph following the discussion of Hirschfeldt and Royer