Any graph of large chromatic number has many edge-disjoint cycles on one subset of vertices

In [1], Erdös and Hajnal raised the following question:

Problem

Is it true that for every \(k\), there is an \(F(k)\) so that if a graph \(G\) has chromatic number at least \(F(k)\), then it always contains \(k\) edge-disjoint cycles on the same set of vertices?

Bibliography
1 P. Erdös, Some recent problems and results in graph theory, Discrete Math. 164 (1997) no. 1, 93–96.