Critical-curve continuation conjecture for the gamma-2 family

Let $\gamma_2$ be the triple Copenhagen homoclinic family, let $\mathfrak{D}$ be the critical curve, and let $L_0$ and $L_2$ be the relevant libration points. Gamma-2 critical-curve conjecture. The $\gamma_2$ family always continues to $\mathfrak{D}$, where it vanishes with $L_0$ and $L_2$ in the saddle-node bifurcation. Numerical continuation and normal-form calculations support this proposed global behavior, but the claim is not established rigorously.