An $n(\log n{)}^{o(1)}$ bound for nested cycles without geometric crossings

Cycles $C_1,\dots ,C_k$ in a graph are called nested without geometric crossings if they are pairwise edge-disjoint, $V(C_k)\subseteq \cdots \subseteq V(C_1)$, and each pair of consecutive cycles induces the same cyclic order on the vertices of the inner cycle, up to reversal. Let $f_k(n)$ be the least number of edges that forces such a family in every $n$-vertex graph. Answering a question of Erdős for two cycles, Gil Fernández, Kim, Kim and Liu proved that $f_2(n)=O(n)$ and asked whether $f_k(n)=O_k(n)$ for every fixed $k$. Xu, Zeng and Zhang recently obtained the first general bound, $f_k(n)=O_k(n(\log n{)}^{k-1}(\log \log n{)}^{k-3})$ for every fixed $k\ge 3$. We prove that, for every fixed $k\ge 3$, $$f_k(n)=O_k\text{\hspace{0.17em}⁣}\left(n\frac{(\log \log n{)}^2}{\log \log \log n}\right),$$ so in particular $f_k(n)\le n(\log n{)}^{o(1)}$, where the $n$-dependent iterated-logarithmic factor has the same form for every fixed number of cycles.