The conjecture on triple-collision shape curves and their first eclipses

Let $\Gamma^{\ast}(s)$, $s\geq 0$, denote a triple-collision shape curve, and let the equator circle be the locus representing the first eclipse. Triple-collision shape-curve conjecture. The different triple-collision shape curves $\Gamma^{\ast}(s)$, $s\geq 0$, intersect the equator circle for the first time at different points; moreover, each point on the circle is reached by a unique curve. The curves do not intersect one another, except possibly after the first eclipse. This is proposed as an open problem because a stronger version of the surrounding geometric theorem appears close, but its proof requires further elaboration.