We have found that the fix is this:
First get the storagemaximumlevel for Central Administration using these PowerShell commands (insert your Central Admin URL):
$ca = get-spsite -identity http://Servername:CAport
A numeric result will be returned.
Then set a bigger maximum for it using the same command but with a number bigger than the result.
No IISRESET is required after running the script.
Updating the Storage Maximum Level is known to resolve the issue.
Items currently in the mail “drop-box” will start processing as soon as the timer job starts.
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