It should be considered open territory imho, the navigation lanes should not be controlled by any single country as they are not "in" any country.
Does the US charge for ships transiting our territorial waters?
No, the United States does not charge ships simply for transiting its territorial waters (the 12-nautical-mile territorial sea). Under international law, which the US follows as customary law, foreign ships have the right of innocent passage. This allows them to navigate continuously and expeditiously through the territorial sea without stopping or engaging in activities that could harm the coastal state’s peace, good order, or security. Coastal states like the US cannot levy charges or fees solely for this passage.