Analytic-conjugacy rigidity conjecture for three-scatterer billiards

Let a $3$-scatterer dispersing billiard system be a billiard system with three dispersing scatterers, and let its collision map describe successive collisions. Two such systems are analytically conjugate near the homoclinic orbit if their collision maps are analytically conjugate in a neighborhood of that orbit.

Analytic-conjugacy rigidity conjecture. Any two $3$-scatterer systems that are analytically conjugate to one another in a neighborhood of the homoclinic orbit must be isometric. Equivalently, given any normal form $N$ and gluing map $G$, there is at most one triple of functions $(\Phi_1,\Phi_2,\Phi_-)$ that can be realized from the collision map of a $3$-scatterer system.

The conjecture would reduce marked length spectrum rigidity for this class of systems to analytic conjugacy. The source explains that proving it is the remaining step in its approach, but gives no resolution.