The map doesn't actually show the optimal path. It is proven that "loops" (the ways crossing each other) can be straightened, and the new version will be shorter. There is one east of Alberquerque.
If I understand Concorde's claims correctly, there is still the question of finite numerical precision (it doesn't seem to use MPFR or any other arbitrary precision library). Perhaps the suboptimality of the path is less than 1e-7 or 1e-16 (depending on precision) times the distance between the "looped" cities?
Having said that, one of the authors of Concorde is R. Bixby, a co-author of CPLEX (which was for decades, and may still be, the industry standard LP solver including for branch-and-bound problems). And Chvatal is another very widely regarded LP researcher. So I would take Concorde's claims of optimality at face value (though of course there could be a data input error somewhere).
Edit: Ah, I misunderstood the sense of "loop"; I thought there was a subcircuit (which I believe can be optimal in some cases), but instead there are two segments crossing each other that, per wrk1's comment below, should really be shorter if their destinations were "swapped". Rough Google-maps math suggests that would reduce the distance by ~10 miles out of ~16k, which seems well above numerical precision.
That said, it's also well known that non-OR practitioners have less confidence in our results when there are trivial local suboptimalities, in some cases even when they don't affect the objective function (e.g., off the critical path in a scheduling problem); I've heard of several professionals who pass the output of exact (modulo stopping criteria) methods through stupid local searches just for that reason.