Aye, it will be for sure a game of whomever gets there first and can keep the place has it.
Especially if nobody else has the mean to make a valid claim by reaching the place, unlike for Antarctica which pretty much every maritime country could reach by the end of the 19th century.
On the contrary, as these places gets more and more reachable and exploitable, I get the feeling that the
Law of the Sea and the
Antarctic Treaty System will soon fall into oblivion.