Nested cycles without crossings — k-cycle version

Establish the existence of a function d_0(k) such that, for every integer k ≥ 3, every graph G with average degree at least d_0(k) contains a sequence C_1, …, C_k of nested cycles with no crossings (i.e., the cyclic order of C_{i+1} respects that of C_i).

Background

Gil Fernández, Kim, Kim, and Liu proved the existence of two nested cycles without crossings for sufficiently large average degree. Extending this to k nested cycles would generalise Erdős’s nested-cycle phenomena to longer sequences while preserving the no-crossing structure.