This isn't unsolvable. In our limited question, most of the physiological factors are constant so can be ignored. Likewise, I can't see how residual gas would make a difference. You are starting with the same percentages of N2 either way and they are eliminated at the rate (again by percentage). So the PPN2 will match breath for breath.Then I think you should screw the physics and look at the physiology instead: the lungs don't fully deflate. Therefore right after gas switch there is a residual air in there that mixes with inspired O2, goes into the tissues, yadda yadda yadda. So there is a non-zero fraction of N2 in there that gets smaller with every breath but is nevertheless there, and is subject to laws of our simple model. QED. (Edit: I mean: you do off-gas faster at 6m, QED.)
Do you know the answer to the physics question?