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You raise a couple of points. For convenience in replying let me number them, as I find this helps me keep track of things.
1. Approximate paths. In the CH there is nothing except Hilbert space and Hilbert subspaces at the fundamental level, and in Hilbert space there are no mathematically precise positions and thus no mathematically precise paths. However, as noted in every textbook it is in many circumstances possible to have paths which are pretty-well defined if one keeps the usual uncertainty relations in mind, and that is the approach that I took in my paper. For the example of interest, in which one can think of the distances traversed by the particle as macroscopic, the sort of approximation I made are the usual sort of thing employed in theoretical physics, so I confess I don’t see why you are making an issue of it. The failure of Bohmian mechanics to give a physically plausible answer is a matter of centimeters, not nanometers. Would you agree?
2. Approximate ontology. My answer to your question, ‘Are you ready to accept an approximate precision in the definition of an ontology?’, is that the CH ontology  is based on Hilbert subspaces, and if for treating certain situations it is useful to use approximations to these subspaces I see no harm in doing so. While it is very useful to employ precise mathematical models in physics in order to keep our thinking straight, it is also a good idea to keep in mind that no theory in physics is exact. We do not know the exact form of the general relatistic corrections to the spectrum of a hydrogen atom, nor even how to do an exact (non-perturbative) treatment in QED.
3. Your Afshar attachment. I was not aware of your article and have added it to my long list of references on that topic. It may interest you to know that early on when he was trying to convince people that there was really something very exciting, he flew to Pittsburgh in order to get my opinion. I am afraid I disappointed him; I told him how to analyze it using histories, but he was not very interested.
Best wishes, and I hope the weather is a bit more comfortable. Bob Griffiths
 R. B. Griffiths, “A Consistent Quantum Ontology”, Stud. Hist. Phil. Mod. Phys. 44 (2013) 93; arXiv:1105.3932