Transverse homoclinic connection conjecture for periodic points $p_n$

Let $\phi_\lambda$ be the billiard map, let $\Omega_\lambda$ be its nonwandering set, let $P$ be the parabolic attractor, let $p_\lambda$ be the fixed point, and let $p_n$ be the periodic points. Transverse homoclinic connection conjecture for $p_n$. For every $\lambda_2<\lambda<1$ and every periodic point

$$x\in\Omega_\lambda\setminus\bigl(P\cap\{p_\lambda\}\bigr),$$

there exist $n,m\geq1$ such that $W^u(x)$ and $W^s(p_n)$ intersect transversally, and $W^u(p_m)$ and $W^s(x)$ intersect transversally. This is the manifold-intersection version of the proposed homoclinic-class decomposition in the range $\lambda_2<\lambda<1$; it remains open in the source.