I wouldn't refrigerate TF-4. It'll form thick precipitate (which you already see at room temp) and it'll be a pain to handle the solution after refrigeration. I did not design or manufacture TF-4 so I won't guarantee anything, but I would guess that TF-4 should keep reasonably well in absence of air at 85˚F. However, as a general rule, chemicals keep longer at a lower temperature.
Bulk chemicals are usually safe to refrigerate, as long as they are tightly sealed, and avoid condensation when opening them.
I store certain chemicals in fridge, but most of them are stored at ambient temp, which can get 85˚F when AC is off.
Incidentally, there is another archival, alkaline rapid fixer on market now, and it is free of sediment at the bottom of the concentrate solution.