Hmmm....
That could be part of the issue right there. There ought to be a way to delete those inbox files, especially if you aren't using them. How many are in there, by chance?
Also, do you have a folder in your public_html directory called "alertlog"? If so, see what is in there, and if there is anything, delete it, and see if that makes any difference.
In that screen shot, there is a folder called logs. Look and see if there are files/folders in there.