All the quantitative measurements show that at 200mm there is nothing to choose between the 70-200mm f/4 IS and the f/2.8 IS II. At 70mm, the f/2.8 is nigh perfect sharpness at the centre whereas the f/4 is very good/excellent. On Sunday, I AFMAd my f/4 with the 2xTC III. At 400mm with a real life target (a very exciting brick wall), it wasn't much different from my 100-400 L on the 5DIII. But, everything is relative when discussing sharpness. The f/2.8 300mm II is so much sharper that it blows the the two zooms away - you can rezz it from 300 to 400mm in PS and it is still far, far better at picking out fine details and contrast.
5D IV, 5DS R, 7D II, EOS-M, Powershot G3 X, 400mm DO II, 300/2.8 II, 1.4xTC III, 2xTC III, EF 1.8 STM, EF 70-200/4 IS, EF 24-105, 15-85, 100-400 II, Sigma 10-20, Sigma 150-600mm C, EOS-M 18-55, f/2 22, 11-22