I just saw the film yesterday and it seems like they have a very half baked theory to the entire time travel plot.
Why can only Agent J remember K correctly?
As in - why can only he remember him and all episodes until the previous day when K gets killed in the past? Agent O says something to the effect of that you knew him before (which also gets revealed in the last scene) - but so did Agent O and I am sure many other people.
So why does only J remember him so distinctly?