While you can't delete a conversation from your LinkedIn Recruiter inbox, you can archive message conversations from within your inbox. If you archive a candidate's profile in a project, any messages will still be available in your Inbox. The archive feature allows you to set aside the messages and retrieve them at a later time.
To archive a message:
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Move your cursor over the Messages icon at the top of your Recruiter homepage and click View all messages.
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Move your cursor over the message you want to archive.
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Click the Archive conversation icon located next to the
Mark as unread icon.
To unarchive a message:
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Move your cursor over the Messages icon at the top of your Recruiter homepage and click View all messages.
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Click the Archived folder on the left pane.
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Click the Unarchive conversation icon located next to the
Mark as unread icon to move the conversation back to your Inbox.