Commutating ferrite ring will saturate more quickly when load current increases...
The ring saturates, but because the frequency and mainly the current slope gets higher and amplitude increase the base drive (the voltage across the ring gets way higher, increasing the base drive way beyond what the transistor needs for the currents, so causes the Ts to go way higher), the quite long storage time in the transistors makes the current to go way larger. And this effect causes the current to go way beyond the normal operating current.
It is not uncommon to see currents 3x higher than normal, but more like 5x...
In reality the opposite is more of a problem: To keep the current from soaring really high, saturating the main inductor and frying the transistors, mainly when the lamp ignition fails (normally the high current rise is stopped by the discharge clamping the voltage across the resonant capacitor).
Also without this much of current boost, it won't be able to generate sufficient voltage for discharge ignition.
Taking an example with a 11W, 0.15A lamp, series inductor about 3mH, resonant capacitor 2.2nF, neglect the series coupling capacitor for the simplicity (it makes the things even worse, as it efectively reduces the inductance), fed from rectified 230V so about 300VDC so about 150VAC at the halfbridge.
Normal operating frequency would be about 36..40kHz, yielding the 0.15Arms (L/R time constant fed by a square wave).
A starting frequency around 140kHz would yield about 0.45A and barely 220Vrms across the capacitor (Vstart = Ires * (1/(2*Pi*Freq); 150V/Ires = 2*Pi*Freq*L - 1/(2*Pi*Freq*C); assume operating on the inductive resonance side). And that is already quite low, even when speaking about 140kHz.
The 140kHz means 3.5us long pulses, only the second half of it is carried by the transistor itself (the first comes from the parallel diode, because it flows in the opposite direction) so about 1.75us while the Ts of the transistors use to be in the 1.5us ballpark. So the ring core saturation needs to happen about 250ns after the current zero cross, which corresponds to about 0.15A saturation current it would have for normal operation.