That is really one of the main advantages of IMAP accounts. If you want to have local storage, use POP3 instead of IMAP.
I have a work-around for you. create a NEW account somewhere as POP3, then any email you want to store on your phone, "forward" it to yourself in this other account. Add this POP3 account to AquaMail. Then you have a sort of "backup" local storage account you can store things in.
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