I'm probably being pedantic but proprioception is derived from touch and sight, and not really a sense of its own. In med school I rotated through neuro clinics and frequently tested for diabetics' proprioception. Diabetics lose this in severe diabetes as diabetic retinopathy and/or peripheral neuropathy set in. Your inner ear fluid also factors in (the dizziness you get from spinning) and of course the areas of the brain that process this information.

Similarly I don't think we call depth perception a sense because it's a derived piece of info your brain processes, or sensation of heat/cold because it's derived from touch.

Basically you use sight, your inner ear (I believe it's specifically the positioning of fluid due to gravity), and the feeling of having your feet on the ground, as inputs for your cerebellum to keep your balance.

Loss of 2 of these things will lead you to not have enough information to derive balance. A simple test for peripheral neuropathy in diabetics: have them close their eyes with their feet together, and give them a very slight tap on the shoulders (while ready to immediately catch them). Diabetics with peripheral neuropathy can't balance well if they close their eyes or if their vision is bad enough, because they've lost 2 of the 3 inputs (vision, and sensation in their feet).