Existence of colliding action minimizers in the spatial N-body problem

Let $N>3$, let $\omega\in(0,N)$, and let $\xi\in\Xi_N$. Let $q^{\omega}$ be an action minimizer of $\mathcal{A}_{\omega}$ among all loops in $\Lambda^{D_N}_{\xi}$ satisfying the coercivity condition in equation (coercive z). Collision-minimizer conjecture. There exist $N>3$, $\omega\in(0,N)$, and $\xi\in\Xi_N$ such that every such action minimizer $q^{\omega}$ contains at least one collision. The preceding theorem restricts possible collision times and collision types, but the authors cannot prove that minimizers are always collision-free and expect that collisions occur for some parameters.