The size of the poset of compatible locally quasi-convex topologies on locally compact abelian groups

For a Hausdorff locally quasi-convex abelian group $G$, let $\text{C}(G)$ be the poset of all Hausdorff locally quasi-convex group topologies on its underlying group having the same continuous characters as $G$. We prove that, whenever $G$ is non-precompact, $\text{C}(G)$ contains an order-isomorphic copy of $(\text{Pow}(\text{cont}),\subseteq )$, where $\text{cont}=2^{{\mathrm{\aleph }}_0}$. The embedding takes values between the Bohr topology and the original topology. Consequently, every infinite discrete abelian group $D$ satisfies $\mathrm{\mid }\text{C}(D)\mathrm{\mid }=\text{width}\text{C}(D)=2^{2^{\mathrm{\mid }D\mathrm{\mid }}}$. The discrete reduction for locally compact abelian groups then yields exact cardinality and width formulas for all such groups; in particular, both invariants equal $2^{\text{cont}}$ for every non-compact $\sigma$-compact locally compact abelian group. These results answer Questions 6.1–6.3 and 6.5–6.7, and the locally compact case of Question 6.4, posed by L.~Außenhofer and D.~Dikranjan in \cite{AD20}. The embedding also applies to non-compact complete metrizable locally quasi-convex groups. A non-compact precompact nuclear group with a unique compatible topology shows that non-compactness alone does not suffice in the nuclear setting. Finally, the compatible poset of ${\mathbb{R}}^{\mathbb{N}}$ is not order-isomorphic to that of any discrete abelian group.