I think you should accept Baker and Howland instead of only accepting Howland and Baker. Also, I believe that "V" comes before "W" alphabetically, so Vanuatu should be higher up your solution list.
The title is wrong - the answers to the quiz are the 'political units' in Oceania - ie the independent countries and dependent territories, rather than islands or groups of islands - but Easter Island and Kingman Reef are missing....
https://en.wikipedia.org/wiki/Polynesia#/media/File:Pacific_Culture_Areas.jpg
Easter Island/Rapa Nui?