Determine the inverse-sliding-map images of the frozen NILP branches

Determine whether the northeast and northwest arctic branches of the non-intersecting lattice-path configurations associated with macroscopic frozen exit regions map, under the inverse sliding map to touching paths, to vertical segments.

Background

Section 3.3 studies touching-path boundary conditions with macroscopic numbers of paths exiting at selected points. Under the sliding map, these multiple exit points correspond to frozen segments in the non-intersecting lattice-path model. The paper derives the southwest and southeast branches but does not obtain the northeast and northwest branches for the touching paths. In the displayed example, the authors formulate the unresolved geometric behavior of these remaining branches as a conjecture: they may collapse to vertical segments under the inverse, or “squeezing,” map.