General locally quasi-convex case of Question 6.4 and broader extension results

Determine whether the cardinality, width, product, and related compatible-topology conclusions sought in Question 6.4 extend from locally compact abelian groups to arbitrary locally quasi-convex abelian groups, and establish the full range of extensions requested in Question 6.8 beyond the cases treated in the paper.

Background

The paper resolves the locally compact case of Question 6.4 and proves substantial extensions for non-compact complete metrizable locally quasi-convex groups. It also gives counterexamples showing that several conclusions do not extend to all non-compact nuclear groups and that the discrete order-isomorphism reduction can fail outside the locally compact setting.

Despite these results, the authors explicitly leave unresolved the general locally quasi-convex version of Question 6.4 and the complete range of extension problems posed in Question 6.8. The statement concerns the unresolved scope of the cited questions rather than a problem solved elsewhere in the paper.