...however, I'm seeing a bunch of nonsense with very few images to back up the claim of questionable performance. I am going to offer an address where I feel you can get the straight scoop.
I have seen his tests of shots he did of the WV capitol handheld with the 100-400 and with stacked Canon "L" extenders, the 1.4x and the 2.0x and I thought they were very very sharp.
If one's copy of the 100-400 is very very sharp, (ie. you have succeeded in cherry picking an excellent, rare copy of the lens) then you would expect to get images when using an extender that are also sharp.
If you are able to use flawless technique, a tripod, MLU when shutter speeds are slow, then it's possible to get a noticeable improvement in clarity and detail by using an extender, even with my rather average copy.
With my lens, I find no benefit at all with a 1.4x extender at the maximum aperture of F8, even when shooting a static subject with tripod and MLU, for reasons I explained earlier, namely, that my lens is noticeably less sharp at F5.6 than at F8 and F11. I believe this level of performance may be typical for this lens.
However, at F11 and F16, with 1.4x extender, I see an improvement in sharpness and clarity at 100% on the screen which, bear in mind, is respresentative of a really huge print, like 5ft x 7 1/2ft.
The problem for me therefore, is that to stand a chance of capturing more detail using an extender, the maximum aperture I can use becomes F11, not F8.
Furthermore, to be certain of capturing that greater detail, one may have to use a faster shutter speed than one would use with the 400mm at F8. A tripod doesn't always solve the problem because any slight movement of subject is magnified as a result of the greater focal length.
The following 100% crops were taken in ideal conditions, tripod, MLU, and completely still subject. As you should be able to see, at both F11 and F16, the text in the 560mm crops are clearer than in the 400mm shots uprezzed. But the 560mm shot at F8 is no better, or hardly better.
In real world conditions when the subject is often moving and one struggles to get a sufficiently fast shutter speed even at F8, it doesn't seem practicable to use an extender, especially if one also loses autofocus.