Persistence of saddle-center homoclinics in the planar elliptic restricted three-body problem

Let $e>0$ be sufficiently small, let $\tau$ be the time variable, and let $R_{e,\mu}$ be the Hamiltonian of the planar elliptic restricted three-body problem, with $L_1$ denoting the periodic orbit that continues the circular-problem equilibrium. The time-$2\pi$ map is taken on the four-dimensional cross-section $\{\tau=0\}$.

Persistence conjecture. For every small $e>0$ there exists $\mu(e)$ for which the orbit $L_1$ has a homoclinic orbit. Moreover, there is a neighborhood $U$ of the curve $\mu=\mu(e)$ such that for each $(e,\mu)\in U$ the time-$2\pi$ map of $R_{e,\mu}$ has a symplectic blender, and the orbit $L_1$ has a homoclinic orbit for a dense subset of $U$.

This predicts persistence of the homoclinic orbit from the circular restricted three-body problem under small elliptic perturbations, together with the occurrence of symplectic blenders and homoclinic connections on a dense parameter subset. The supplied text presents this as reasonable to believe, and no resolution is given.