Mayo is made from raw eggs so iis a high risk food item

However as pointed out "most" branded mayo as egg powder so much lower risk as the protein content is lower which is what bacteria grow wioth

SO to be safe store in the fridge

Pickles and Jams you can keep in your cupboard once opened as the protein is VERY low and the sugar /Vinegar content means less water moisture therefore low bacterial growth rate

Hope this helps

NIge