\[fig:dispersion\] of the main text. A list of definitions is given in Tab. \[tab:defs\], and the fit parameters are given in Tab. \[tab:fit\]. For all measured QDs, the strong magnetic field dispersion of the radiative Auger emission lines is excellently reproduced by this description.

In the case of the InGaAs QD shown in Fig. \[fig:dispersion\](a,b) (main text), we fit the energies of the $s$-shell emission and the radiative Auger emission into both $p$-shells simultaneously. The coupling term $\Delta_{pd}$ is included as a fit parameter. The exciton $g$-factor is measured independently by mapping out the charge plateau of the trion in a magnetic field. The fit reproduces the data very well and gives a good description of the radiative Auger excitation into some of the $d$-shells. The LO-phonon resonance of the radiative Auger excitation into the $p$-shells is also excellently reproduced by the fit.

To fit the magnetic field dispersion of the InGaAs QD shown in Fig. \[fig:mechanismFull\](b), we also make a simultaneous fit to the energies of the $s$-shell emission and the radiative Auger emission into both $p$-shells. The coupling term $\Delta_{pd}$ is not included as there is no hint of an anticrossing with the $d_+$-shell.

For the GaAs QD 