Yes, fixer will die because of oxidation.
It's too bad that the TF-5 data sheet doesn't have more detailed information on the life of the working solution in trays, half-full bottles, etc. What they do say is:
"STORING TF-5
All photographic solutions should be used as soon as possible. TF-5 stock solution should not be kept for more than 12 months. All fixers should be kept away from high heat (over 85° F), as this will shorten the shelf life. If TF-5 is to be kept for some weeks after opening, it should be placed in smaller bottles filled to the top, with tight caps."
My added emphasis indicates to me that TF-5 may be more prone to oxidation than other fixers. Too bad Ron Mowrey is no longer with us to answer your question and provide some more information.
FWIW, Ilford states that their rapid fixer working solutions will last only 7 days in an open tray and only one month in a half-full bottle. TF-5 may be less than that.
Best,
Doremus