Then brace yourself for the disappointment then, because a 400/2.8 with a built-in TC wouldn't be a "Mk III" - it would be the first version of such a lens.
It's not that I don't understand why you want it, it's just that this is not what the rumor is telling.
The keyword here is "rumor".
And I will for sure not take anything for granted until we receive official information on this topic.
Canon introduced the built-in TC in their 200-400 that was released in 2013.
(After they changed the construction. They shoved a prototype in 2011).
The 400/2,8L IS II USM was released in 2011.
To me, it would make no sense at all if they wouldn't have worked on 400/2,8 with a built-in TC.
Let's remember that there is a seven-year timespan since the release of the 400/2,8L IS II USM.
I might end up being totally wrong here.
But it would make no sense to me at all.
If Canon wouldn´t use their knowledge from developing their 200-400/4 and built-in TC, with the upcoming 400/2,8.
From a business perspective, a built-in TC makes a ton more of sense, than not having it built-in.