The topology of a topological group $G$ is called an ${\cal L}_{\infty}$-topology if it can be represented as the intersection of a decreasing sequence of locally compact Hausdorff group topolgies on $G$. If ${\cal L}_1 < {\cal L}_2$ are two distinct ${\cal L}_{\infty}$-topologies on an Abelian group $G$, it is shown that the quotient of the corresponding character groups has cardinality ${\geqslant} 2^{\rm c}$. A conjecture in this sense announced by J. B. Reade in his paper [6] is thereby proved.