Its fairly typical of a Basenji not to want to soil his own domain. I've had many over the years and owners of our puppies have experienced the same thing. NOT at home ! We have about 2/3 of an acre of garden including lawns, borders, shrubbery, orchard - plenty of space, you'd think, for a Basenji to do his/her business. But some never have and never would.
One other Basenji mine walk with (run free with) in the woods goes one better, he won't even pee in his own garden but saves it up for the woods. When he pees, it lasts often for over a minute and can't be good for him. Having pee-d - he then finds a hiding spot and defecates.
In the woods (or out on a more conventional walk) - NEVER at home !