Tout d'abord:
Les TC III n'améliorent pas les perfs des objectifs. Ils dégradent moins par rapport à la version II : nuance!!

Ensuite, voila ce qu'on trouve sur Canonrumors:
Just to clarify, the 70-200 f/2.8L IS II does not benefit from the AF precision improvements of the new III series TC's......; I believe development of the 70-200 was too far ahead of the new super-teles/TC's to take advantage of the improvements.
(Recommendation 70-200/2.8+2x vs 100-400 f/4-5.6L ? )

et sur thedigitalpicture:
( Canon EF 1.4x III Extender Review )
Canon EF 300mm f/2.8 L IS II USM Lens,
Canon EF 400mm f/2.8 L IS II USM Lens,
Canon EF 500mm f/4 L IS II USM Lens and
Canon EF 600mm f/4 L IS II USM Lens.
Improvements in the Series III extenders promise to perform especially well on these specific lenses.

A microcomputer integrated into the Series III Extenders promises faster autofocusing and increased AF precision when used with the Canon IS Supertelephoto Series II lenses mentioned above and (I expect) all future compatible lenses. Better optical quality is also expected.

Note that "AF precision remains the same as the Series II Extenders when the Series III Extenders are used with earlier extender-compatible EF lenses." [Canon]