Dual layering of compact abelian group actions

Prove that if a compact group $K$ acting on a compact abelian group $G$ is compactly layered by a decreasing chain of subgroups $(G_n)$, then the dual action of $K$ on the Pontryagin dual group $\widehat{G}$ is discretely layered by the subgroups $F_n=\widehat{G_n}$.

Background

The paper proves the asserted correspondence for the action of $GL_d(R)$ on $R^d$, where the compact action is layered by the subgroups $M_d^n$. It then seeks a general result relating compact layerings to discrete layerings under Pontryagin duality. A partial proof reduces the issue to showing that a certain induced action on the first dual layer is transitive, but the final step is explicitly stated to remain unresolved.