Hi, I have just returned from Rarotonga and have viewed information on Wigmore & Loschmann in the Rarotonga records.
William John Wigmore had 2 children to Emilie Loschmann in Tahiti before his return to Rarotonga, He leased land in Rarotonga where his family can still be found if you search the Mormon Microfiche for the period of 1916 you will find this infromation including a document from William & Emilie. I hope this helps your search.
I am trying to connection Wigmore's in NZ to Rarotonga.