Homoclinic relation conjecture for the periodic points $q_n$

Let $q_n$ be the periodic points of the billiard map, let $W^s(q_n)$ and $W^u(q_n)$ denote their stable and unstable manifolds, and let $c_n$ and $\lambda_1$ be the bifurcation parameters used in the paper. Homoclinic relation conjecture for $q_n$. The manifolds $W^u(q_{n+1})$ and $W^s(q_n)$ intersect transversally for every $0<\lambda<c_{n+1}$. Moreover, if $0<\lambda<\lambda_1$, all $q_n$ are mutually homoclinically related for sufficiently large $n$. This is a numerical conjecture about the invariant-manifold structure of the periodic points; no proof or resolution is supplied.