The Basenji breeders certainly believe their dog is an ancient one. Greyhound fanciers will proudly tell you their dogs originated in AE. Greyhounds figure prominently in Medieval texts & in the surviving art. Most dog fanciers believe that the Greyhound is the oldest 'pure bred', with a few championing the Saluki. Certainly the evidence from AE bears this out. I don't think it's a coincidence that the oldest known breeds come from the ANE, nor that they are the swift sighthounds.

The breed people do a tremendous amount of research on the origins of their beloved animals. I'm part of the Poodle History Project, for example, and there are people in that group who have pored over Medieval texts to glean any information about dog types. My contribution is to hunt through archives of various quality art works in search of dogs that are, or could be, Poodles. Thus far the earliest definite 'proto-Poodle' is from @1100 AD.

The extremely tight-curled tail on some of the AE dogs is identical to the tail on the modern Basenji. The dog shown chewing on the ankle of the gazelle is definitely acting like a Basenji.