Its stopped snowing, and the 1 inch of snow has melted, so I went out to the studio and pulled out my 70-200 f/4L IS and 100-400mm L and taped the contacts on the lens rather than the TC, simply because it was temporary and easier.
First, I tried the 100-400mmL and 1.4X TC (f/8 equivalent). At 100mm which is a little faster than f/8, the lens did focus fairly reliably but not fast. At 400mm, it hunted, but did manage to focus. I suppose its usable, but pretty slow. I panned the lens at 400mm from distant to near, and the focus did follow fairly well.
Then, I put the 2X TC on my 100-400 (f/11) and it tried and erratically found focus, but I'd call it a failure, because it sometimes focused incorrectly (fuzzy image).
Next, I taped the 70-200mm f/4 L IS and attached the 2X TC (F/8). I tried it at 70 and 200mm as well.
At 70mm, the lens was unstable, but eventually found focus for near objects. For far objects, it focused slowly. At 200mm, the lens would not focus on near objects, but wanted to chatter like my 135mm L. At intermediate and far distances, it failed to focus except sporadically, so I call that a failure.
My 70-200mm f/2.8L (Non IS) focused fine, quickly at 70mm and 200mm every time with a 2X and 1.4X TC stacked (f/8 equalivent).
My 100-400mmL + 1.4X TC worked, but not well, and would focus slowly with considerable hunting.
My 70-200mm f/4 L IS was basically unusable with a 1.4X TC.
My 135mm L worked fine at longer distances with stacked TC's, but was slow to focus.
I'd say that a firmware fix would not work for reliable f/8 AF, a new sensor design would be needed, but some combinations of taped TC's and lenses might just work.
I'd be interested to here if the new 70-200mm f/2.8L IS MK II works like my old one. Mine worked suprisingly well.