A government in exile doesn't really count as control, though, so I'd say from 1942 to 1944/45 the islands were "under" Japan, the US and the Commonwealth government retaining their claim but not yet able to enforce it. For a few months during the Japanese invasion and the subsequent US recapture, parts were Japanese-controlled and parts US-controlled, but in between the US was out of the picture in terms of de facto possession.