Perimeter-preserving bijection for two-staircase unimodal polygons

Let a unimodal polygon have two staircase regions with identical sides, possibly differing widths, and let the two boxes delimiting the staircase region be identified as in the proposition. Consider the polygons whose bottom perimeter intersects the right box but not the left, and those whose bottom perimeter intersects both boxes. Perimeter-preserving bijection conjecture. There exists a perimeter-preserving bijection between these two classes of polygons, where perimeter preserving means that the horizontal and vertical perimeters remain constant. The proposition establishes equality of their generating functions, but the existence of an explicit perimeter-preserving bijection is not established here.