As the 'ballast' accepts a range of voltages, I assume it is an electronic 'lamp power supply', i.e. an SMPSU rather than a wirewound ballast. This is an important distinction. An LPSU runs at constant power and will normally drive the lamp correctly at full power or not at all. The lamp would not be underrun, as it would be with a 120V wirewound ballast. However, when the supply voltage is too low, the input current will be increased and at some voltage, the unit will shut down to protect itself. Or it should; there is a possibility of blowing its internal fuse in the meantime.
Most 120V equipment will work in Japan so long as the voltage is actually around 100V and not affected too much by drop in the wiring; 85V might be pushing your luck. It will depend on how well designed and built the LPSU is.
Many years ago I remember a new and very expensive piece of kit that kept shutting down due to undervoltage when we were using it in Tokyo on its 120V setting. In a bit of a hurry, I nicked the transformer out of something else less important and reconnected it as an autotransformer, with part of the secondary working as a boost winding to make 120V from 90.