If they were deleted, they aren't there for any e-mail client to
download. Use the webmail interface to your e-mail account to look at
your mailbox. If the mails are there, they weren't deleted. If you
don't want to download all the mails there, don't keep using your Inbox
for permanent storage of old items. In the webmail interface, create a
new holding folder and move the mails there that you don't your e-mail
client to see in your mailbox (aka Inbox).
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