Openness of the inclusion in LB-completions

Determine whether there exists an LB-space for which an LB-completion exists but the canonical inclusion i into its LB-completion is not open onto its range.

Background

The notion of LB-completion is defined via a universal property within the category of complete LB-spaces. When the usual completion \widehat{X} is LB, the inclusion is open onto its range. It is unknown whether there can be LB-completions where this openness fails, which would affect the behavior of cokernels and exact structures.

References

We do not know if there is an LB-space for which an LB-completion exists and $i$ is not open onto its range.

— 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 Definition 5.1 (DFN-LB-COMP)