I think Huw has it right. This has always been my understanding of how the sync works in a leaf shutter.

In any case one only needs speeds faster than 1/125 s. when ambient light levels are very high, as in bright sunlight, or if one were using an extremely low power flash, perhaps to stop action. With normal strobes in ordinary room light all at conventional levels, it would be unusual that the strobes would not be at least four stops brighter than ambient.