Conjecture on the caustics of iterates of off-center reflections

Let $R_r$ be the off-center reflection map with parameter $r$, let $m$ be a positive integer, and let the caustic of an iterate be the associated envelope curve. The 2-periodic points of $R_r$ are $0$, $\pi$, and the two points $\pm\phi_c$, where $\phi_c\in(0,\pi)$ satisfies

$$\cos\phi_c=\dfrac{1-\sqrt{1+8r^2}}{4r}.$$

Caustics of iterates conjecture. For $0<r\leq 1/3$, the caustic of $R_r^{2m+1}$ is a $C^\infty$ curve with exactly four cusp singularities, with two of them occurring at $\phi=0,\pi$. On the other hand, the caustic of $R_r^{2m}$ is a differentiable curve, $C^\infty$ everywhere except at exactly the four 2-periodic points of $R_r$, where the caustic is tangent to the unit circle. This conjecture gives the exact caustic structure in the parameter range where the iterates remain diffeomorphisms; the preceding theorem proves only that odd iterates have at least four cusps for sufficiently small $r$, while the stated exact description and the even-iterate regularity remain to be established.