Separating-parameter conjecture for the gamma-3 family

Let $\gamma_3$ be the homoclinic family, let $\mathfrak{D}$ be the critical curve, and let $m_1,m_2,m_3$ be the mass parameters. Gamma-3 separation conjecture. The $\gamma_3$ family continues to $\mathfrak{D}$ when $m_2\approx m_3$ and does not reach $\mathfrak{D}$ when $m_1\approx m_2$; there is a single parameter value on $\mathfrak{D}$ separating these behaviors. The conjecture summarizes the symmetry-based and numerical evidence for the two different continuation regimes.