Please don't give advice that may pose danger to people's health.
The USDA recommends refrigeration for OPENED mayo.
The mayo itself is supposed to not spoil if left unrefrigerated after opening ONLY IF NO BACTERIA IS INTRODUCED INTO IT.
That is rather hard to guarantee when cooking. Just some food on the knife or spoon you use or from other foods while preparing could enter the jar and then grow and be very dangerous.
Refrigeration of mayo is just a good safeguard. It also protects the flavor, which will degrade quickly after opening, if not in the fridge.