I think that buoyancy is not related to depth, yes the pressure on the object will increase overall at greater depth. However, the differential pressure (upward force - downward force) will remain unaffected. The weight difference is insignificant from the law of universal gravitation...
I believe that I am making this a bit harder than it should be.
An object is weighted so that it "hoovers" 5 feet underwater. It is neutrally buoyant. A diver drags the object even deeper without adding weight, she releases the object at a deepth of 100ft. Will the object stay put, rise up...