It is good to keep it refrigerated from a quality standpoint, it's just not required from a safety standpoint.
Form the linked fact sheet:
Q.What happens if I leave
for a long period of time?
A. From a food safety standpoint, commercial mayonnaise and
mayonnaise-type dressings are perfectly stable when stored at room
temperature after opening. Quality, not safety, is the only reason
the labels on these products suggest that they be refrigerated after
opening. Refrigeration ensures that the commercial mayonnaise
keeps its fresh flavor for a longer period of time.
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