Question about Nikon AF-S Nikkor 70-200mm f/2.8E FL ED VR

I have this lens and the TC-20E III, and they are perfectly compatible, and even amazing in terms of image quality.

According to research, the products should fit together https://www.nikon.ch/de_CH...

I checked with Nikon Switzerland, I was assured that it is compatible (and also the autofocus works).