I plan to use the teleconverter on a 400mm f2.8L IS II and 1DX, 5DIII, etc.
I shoot a lot of sports, so fast and accurate focus is important. You don't often get a second chance. The speed and accuracy is supposed to favor the 1.4x III. I've seen the claims that it's faster, but no numbers to back them up and the 1.4x II works quite well.
For this particular application I am not too concerned about the corners. Sharper corners is better, but not a priority.
To my eyes, looking at the center and mid-frame, this comparison on tdp favors the 1.4x II, but they are close. Unfortunately, they don't have a comparison, using the 400mm. I guess it could be my eyes, or maybe tdp the setup needed a bit of AFMA. That's why I am hoping to find more examples.
$200 is not a lot of money in a relative sense and I'll probably bite on the 1.4x III, but am just trying to make an informed decision, before I commit.
See now this is why I always say that I take TDP with a big grain of salt. They have the 1.4x III clearly looking worse wide open on that lens, but that sure does not match the findings of most others. At worst, it should merely look the same, not so clearly worse in the center and middle of the frame.
Anyway Canon says the III offers improves AF precision used with 400 2.8 II. I can't recall if that was for one shot only for for AI Servo too.