I own both lenses and 2x and 1.4x teleconverters. Comparing tons of photos that I've done to manually check lens calibration/AFMA between them with a Spyder LensCal, with, without the TC's, whatever -- my copy of the 100-400LII is sharper at f/5.6 than my copy of the 70-200/2.8 IS II for everything except close to the wide end (near 100mm). This is also true of the chromatic aberration.
The consistency of focus from Reikan FoCal is also a little better on the 100-400, but since both are excellent, that's neither here nor there.
I don't think this is an isolated copy thing, because I borrowed (a different copy) both of those lenses before I purchased either, and I have a couple of friends with both of them too, and we've all commented on how clean you can pixel peep the 100-400LII.
But it's neither here nor there. If you own both lenses, there is no way you'll stick a 2x TC onto a 70-200 to use a 400/5.6 unless you happen to be somewhere without the 100-400. The extra collapsed length, greater weight, loss of 100-140, and greater MFD all just make the 100-400LII a better tool for that job.
I would really welcome a 70-200/2.8 that had the improvements in the 100-400LII, and I would probably spring for it, because this is my cat photography lens. Mode 3 IS would be really nice too, for backyard birding.