Strictness of inclusion between exact and deflation-exact structures on complete LB-spaces

Determine whether, on the category LB of complete LB-spaces, the inclusion of the maximal exact structure E into the maximal deflation-exact structure D is strict; that is, decide whether there exist conflations in D that are not in E.

Background

The paper proves that the inclusion of topologically exact sequences E into the maximal exact structure E is strict, but whether E \subseteq D is strict remains unknown. Resolving this would delineate the gap between exact and deflation-exact structures on complete LB-spaces and refine the classification of acyclic complexes.

References

We have $E\subsetE\subseteqD$. That the first inclusion is strict follows from Example (ii); it is unknown if the second one is strict or not.

— A homological approach to (Grothendieck's) completeness problem for regular LB-spaces  (2512.13161 - Wegner, 15 Dec 2025) in Section “Complete LB-spaces” (SEC-COM), after Theorem LBc-Defl, final remarks