///We don't know that memory becomes static and unchanging. This is a hypothesis that has not been tested.///

It's self-evident. Every memory you have eventually reaches a point where it will not change. Early on it will change. But at a certain point, it becomes set. You recall it over and over and it's the same. Early on, the memory was fresh, vibrant. But as time wears on, details of that memory drop out. Sensations invoked by that memory fade. Whatever confusion you had about the order of the events and the events themselves have long been resolved. Probably not all the accurate but that's beside the point. Think back to a memory you had at 7. Does it change now? No, not unless you remember something that you never before considered but that is virtually never going to happen. The memory is set now--all you remember are certain essential bits because you were conscious of them when they happened. 20 years from now, that memory will still be the same and even if it isn't (it may in fact grow even dimmer) it just means it hasn't reached the stabilizing point yet but it will. It has to.

At the end of time, all memories will have become set and unchanging and all sensations will have dwindled to zero. If that were not the case, it wouldn't be the end of time.