I think it is a theorem in economics that this must happen. Before stating the theorem, we need a definition. I am going to assume that the home nanotech facilities are finite. The cost of retooling or reprogramming a machine is nonzero, and the cost varies depending on what kind of retooling is done. The home nanotech facilities have an interface that requires some expertise and effort on the part of the user.
Different tasks requite different degrees of expertise and effort.
The Gradient Theorem. If the home nanotech facilities are finite, then the MNT-Autarky scenario is not stable. Any approximation to it is unstable. In fact, there is a gradient away from self-sufficiency, in the direction of more specialization and more trade. Informal proof .