Here is a quick link to the website above.
Read the info in italics in the center of the first page of the pdf beginning with the word Commercial.
Scroll down to the second page of the pdf and read the question and answer in the bottom left corner.
Finally read the Q & A starting in the center of the second page of the pdf concerning the "Mayo Myth".
As stated by Bill,"Mayo doesn't need to be refrigerated before its opened as evidenced by grocery store shelves."
In my personal experience and practice only, no recommendations are made or implied here.
At home my mayo is kept in the refrigerator after it is opened. Before opening it is on the shelf in the pantry stored the same way it was in the store.
I am not a fan of refrigerating any food stuff and then putting it in my food bag to be left unrefrigerated for the duration of my hike.
In regards to mayo I was taught from young that it was OK unrefrigerated but once it was in the fridge it should always be put back in the fridge. I'm pushing 60 and back when I was young is quite a while back.;-)
It may be that perhaps what I was taught was due to another "Mayo Myth". That being said, I personally don't shy away from opened and unrefrigerated mayo in the lunch room. I also don't get overly concerned about expiration dates on salad dressings in the lunch room fridge. I also haven't been to the ER with any cases of food poisoning either.
All of the above personal experiences and practice can be taken with a grain of salt or a bicarbonate of soda as needed. ;-)
IMO ask a knowledgeable professional which admittedly I am not.
Err on the side of caution, follow all instructions on the label, never eat anything past its expiration date and if in doubt throw it out!
Above all ask questions, be safe and enjoy your hiking.