Absent any forces, they can only stay on a geodesic. It’s like the arrow that springs from the bow. No hesitation, no doubts. The path is clear. There’s only ever one geodesic, one path through spacetime that takes no energy to follow.
This is true even if you ignore general relativity and just focus on classical mechanics, as formulated by Lagrange. In Lagrangian mechanics, you must compute a number called the “action” for every path through space that your system could possibly take. Then your system will take the path that minimizes the action. If your system is a particle in space, then it will stay still if that has the minimum action, or it will follow some path through space if that has the minimum action. It can’t ever do both. Only one of them will minimize the action.