Equality of maximal exact structures on complete LB-spaces

Ascertain whether, on the category LB of complete LB-spaces, the maximal exact structure E equals the restriction E \cap LB of the maximal exact structure on all LB-spaces; that is, determine if E = E \cap LB holds.

Background

The paper defines several conflation structures on LB and its subcategories (regular and complete LB-spaces), including maximal exact structures obtained from stable kernel-cokernel pairs. Whether the maximal exact structure on complete LB-spaces coincides with the restriction of the maximal exact structure on all LB-spaces is unknown, and would clarify the relationship between these homological frameworks.

References

We have $E\subseteqE\capLB$ and equality is equivalent to the condition in Dfn 2.4(i); it is unknown if this holds.

— 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-MAX-THM, final remarks