Nested cycles with no geometric crossings conjecture

Two edge-disjoint cycles are nested with no geometric crossings when the vertex set of one is contained in that of the other and the two cycles can be drawn together in the plane without crossing edges. Nested-cycles conjecture. For every $k$, there exists $g(k)$ such that every graph with average degree at least $g(k)$ has $k$ nested cycles with no geometric crossings. The source states that the conjecture remains open; only the case of two such cycles is known.