Completeness of the Mackeyfication of LB-spaces

Determine whether the Mackey completion (mackeyfication) X^ of every LB-space X is complete.

Background

The Mackey completion transforms a locally convex space into a Mackey complete one, and for LB-spaces the authors use it to define adjunctions and certain cokernels. However, whether this process always yields a complete LB-space is unknown and impacts structural results for LB-spaces and their exact categories.

References

  1. It is unknown if the mackeyfication $X{}$ of an LB-space is always complete.
— 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), opening paragraph