In order to keep your emails safe, account must always be configured with POP3 settings.
POP3 option download the emails to email client. However, IMAP just shows the email that are under webmail account and not download them.
Once the emails are deleted from webmail sever, emails will not be shown under IMAP settings but not under POP3 settings.
First please configure AOL email account with POP3 settings with apple mail and check if it recovers the emails or not.
AOL does not provide any option to recover lost emails for free account holder.
In this case, please call AOL customer service and ask them if you upgrade then will they be able to recover your the lost emails.
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