This is known as Klyachko's Car Crash Theorem. It was proved in order to prove a theorem about finitely presented groups. In fact, the result allows the ants to move at arbitrary nonzero speeds so long as they make infinitely many loops around their 2-cell. The conclusion is that there's either a collision between ants in the interior of an edge, or else there is a `complete collision', which means that there's a collision at a vertex of all ants from adjacent edges.
EDIT: Oh, it's actually important that there are two complete collisions, which is somewhat harder to prove than one (a collision in the middle of an edge is also a complete collision.)
You can read an expository article by Colin Rourke here.