A 300-600 f/4 would be huge. An f/4.5 slightly smaller but might create problems when used with TC's, especially the 2X, i.e., 600-1200 f/9 wouldn't be usable with 1Dx AF while a 600-1200 f/8 would. A 300-600 f/5.6 wouldn't be usable at all with a 2X TC. The problems are even worse with camera bodies that only AF with lenses f/5.6 or faster.
So, at the risk of repeating myself yet again, why not a 200-500 f/2.8-4, that holds f/2.8 max aperture from 200mm to 350mm? And why not a 300-800 f/4-5.6 that holds f/4 from 300mm to 560mm?