A better kernel for treewidth-t modulator

Treedepth-d modulator as a parameter Somnath Sikdar The recent kernelization algorithms in very general sparse graph classes such as graphs of bounded expansion use the structural parameter of a constant-treedepth modulator [1]. The parameter seems natural in these graph classes, but we have not yet investigated its full power also in general graphs. What natural problems admit a polynomial kernel with respect to this parameter in general graphs? Are there many examples of natural problem that, under this parameterization, admit a polynomial kernel in sparse graph classes, but not in general graphs?