Invariance of the perimeter of focus-inversive polygons

A focus-inversive polygon is obtained from an $N$-periodic billiard polygon by the focus-inversive construction described in the paper. Its perimeter is denoted by $L^\dagger$. Perimeter invariance conjecture. The perimeter of the focus-inversive polygon is invariant for any $N$. This is experimentally observed in the paper; the preceding results establish the analogous invariance for the $N=3$ focus-inversive family, while invariance for arbitrary $N$ remains unresolved.