Again, you are ASSuming.
A VFD is like an entirely new power source for the motor. All it needs from the line side is raw energy to convert to DC, everything else from that point on going to the motor is fabricated by the VFD, and the VFD can fabricate it to be whatever is needed.
That is right but here the point of debate is the effect of input voltage variation on VFD. The VFD has over/under voltage alarm trips to prevent any fault.
But I picked 375V because that is below the threshold of OTHER control components working correctly, such as anything with a coil. So the SYSTEM might trip, but the VFD itself might not.
The system might not trip but the relay contacts would chatter due to low voltage because the drop out voltage for relays are usually below 20%.