Sigma 100-400 + TC-1401 will not AF on 6D

Purchased this combo specifically to use on my trusty Canon 6D, but disappointed to learn that it will not AF. The lens alone works fine. This anomaly was confirmed by Sigma tech rep. Something to the effect that AF on 6D will not work when max aperture exceeds f/8.
 
The 6D, like most older Canon bodies, does not autofocus above f/5.6 (third-parth f/6.3 lenses actually lie to the camera that their max aperture is f/5.6). Many newer bodies AF up to f/8 with the center point, including the 7D2, 80D, and 6D2. Only the 5D4 and the 1DX2 AF at f/8 with all points (but depending on the lens/extender combination, possibly fewer). The Live View AF doesn't have a hard limit and some people have managed to get DPAF to focus up to f/11, albeit slowly, if light is plentiful and there's enough contrast.
 
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