Smooth satellite filtrations with topologically slice images

Let $P$ be a winding-zero pattern whose induced satellite map $P:{\mathcal{C}}_{\mathrm{s}\mathrm{m}}\to {\mathcal{C}}_{\mathrm{s}\mathrm{m}}$ is nonconstant and whose image consists only of topologically slice knots. For $n\ge 0$, let $P^n$ be the $n$-fold composition of this map, with $P^0$ the identity, and let $⟨S⟩$ denote the subgroup generated by $S$. Must every quotient $⟨P^n({\mathcal{C}}_{\mathrm{s}\mathrm{m}})⟩\mathrm{/}⟨P^{n+1}({\mathcal{C}}_{\mathrm{s}\mathrm{m}})⟩$ have infinite rank, where the rank of an abelian group $A$ is ${\dim }_{\mathbb{Q}}(A{\otimes }_{\mathbb{Z}}\mathbb{Q})$? This is the smooth graded-quotient question recorded by Cha and Kim; the question itself supplies no premise for a proof.

Smooth concordance group

Topologically slice knots

Satellite operator and iterates

Cha–Kim, Remark 1.1

Cha–Kim's published 2025 paper still states this question in Remark 1.1. A targeted search on 2026-10-21 found no later resolution notice; this is not an exhaustive literature survey.