How does the 300/4 PF + TC14EIII (or AFS 300/4 + TC14EII, or AFS 80-400VR) compare with the AIS 400/5.6?
If the results are comparable, Nikon management may consider the AF version of this lens already exists in one form or another.
The mass market generally prefers zooms or smaller and more affordable lenses, long slowish primes are rather specialized lenses with a limited market. Sales of the AFS 300/4 totaled about 50,000 units between 2000 and 2015, with at least 12,000 units of the AFS 300/4 PF since then. That is only three times the rate of the various AFS 300/2.8 models over the same time. Nikon have sold over a million 70-300VR lenses. Sales of the AFS 80-400VR is already near 60,000 in less than 3 years, and serial numbers of the 200-500VR suggest production of 25,000 in less than 6 months! By comparison, the slower "affordable" 300/4 lenses are not big sellers, although they sell at a steady rate.
The AFS 400/5.6 would be an even more specialized lens. Compared to the 300/4 it is longer and slower so less easy to use. It's not a zoom and has fewer options for accepting tele-converters so is not a very flexible option. I think Nikon prefers to make telephoto primes with an aperture or f/4 or faster so there are more options for using TCs if you need more reach (the exception is the AFS 800/5.6 but that's at the extreme end of the scale). Nikon sold around 12,500 AFS 400/2.8 lenses of all models since 1998. Assuming a 400/5.6 sells at about three times the rate of the 400/2.8, that would be less than 40,000 units over the same time. Only special purpose lenses like the 200/4 micro or PC lenses sell less, so Nikon probably consider the market is too small.
But that's enough analysis ... such a lens would be nice, and if initial sales of the 300/4 PF and 200-500 are anything to go by, there might be more demand than they realize...