Perhaps I've lost track of what is meant by "two skinny wires". My own
analysis was to calculate at an arbitrary location in space the field that
would be produced by a solid wire, and the field that would be produced
instead by a wire whose diameter is the same as that of the hole and which
carries current in the opposite direction. The currents for these two
different current distributions are adjusted appropriately so that the
vector sum of the two calculated fields is equal to the field at that
location in space for the actual wire that has a (longitudinal) hole in it.
Since the sources are the same, the field must be the same, at all
locations in space. No?