Equality of maximal deflation-exact structures on complete LB-spaces

Determine whether, on the category LB of complete LB-spaces, the maximal (strongly) deflation-exact structure D equals the restriction D \cap LB of the maximal deflation-exact structure on all LB-spaces; equivalently, decide whether in the description of D one may replace “semistable” with “surjective.”

Background

The authors show inclusion D \subseteq D \cap LB for the maximal deflation-exact structure, with semistable cokernels playing a central role. Whether equality holds (i.e., surjective implies semistable in this setting) would impact the characterization of conflations and acyclic complexes in the complete LB category.