There are several ways how a situation like this can be resolved:
(1) If you ever have a certain amount of outpost levels (3? 5?) above the other country, you can just claim the colonial region.
(2) The crisis temperature reaches the critical limit and triggers a crisis over the region that is either decided by diplomatic means with one side folding or during the triggered war.
(3) One side withdraws its colonial efforts in that colonial region (interface button, I doubt that the AI ever does it).
(1) If you ever have a certain amount of outpost levels (3? 5?) above the other country, you can just claim the colonial region.
(2) The crisis temperature reaches the critical limit and triggers a crisis over the region that is either decided by diplomatic means with one side folding or during the triggered war.
(3) One side withdraws its colonial efforts in that colonial region (interface button, I doubt that the AI ever does it).