5 problems

Let $(G,\tau)$ be an abelian topological group, let $G^\wedge$ be its dual group, and consider the locally quasi-convex group topologies on $G$ whose dual group is $G^\wedge$; thes…

Finest precompact topology conjecture. If for $(G,\nu)$ there exists the $\operatorname{MAP}$-Mackey topology associated with $\nu$, then $\nu$ is the finest precompact group topol…

Discreteness conjecture. If $G\in \operatorname{MAP}$ is a Mackey group in $\operatorname{MAP}$, then $G$ is a discrete group.

Extension conjecture. Proposition 5.4(c) should remain true for a not necessarily complete metrizable topological abelian group.

Metrizable LQC Mackey conjecture. Every metrizable $(G,\mu)\in \operatorname{LCS}$ is a Mackey group in $\operatorname{LQC}$.