They continue to decay, and will contaminate clean crops. Onions whose roots alone have been damaged may store, and although they will be small, they will keep well. If you are certain that your compost heap will reach 50 C for several consecutive days, then it is safe to compost contaminated foliage. Otherwise, burn, or landfill, diseased material. When infection is triggered in spring, there will be minimal root damage before higher soil temperatures halt the disease.
Leeks are worth trying, even on really infected ground.
They may survive and crop well due to a combination of soil temperature and root development at a crucial time of growth. Once a white rot sclerotia has germinated and re-infected a new crop, it dies. If you know you have the disease, plant an Allium crop, wait for signs of infection to appear, then immediately lift and dispose of all the crop and surrounding soil. This might be successful in exhausting and removing most of the infection but would have to be repeated more than once. Preliminary tests in pots have indicated that garden compost, when added to soil can suppress diseases such as white rot.