Applications :: Save PDF Attachments From Mail Messages Automatically?

I find this maddening and have searched high and low for help, so hopefully you've got an answer.

In my Mail.app I have two accounts, one for personal email and one for work email. I also have a local mailbox titled "Submitted Forms." I receive forms from coworkers as PDFs in my work email account. I have a Mail rule established so that any email coming into my work email account meeting certain criteria (having an attached form, specified by name) is moved to the local mailbox "Submitted Forms." That all works fine. What I am trying to achieve, ideally using an applescript run by the rule, is to have any message that gets moved into that mailbox save its attachment to the desktop (or other destination folder). I have tried over a dozen similar-but-different example applescripts I've found online that claim to do this, but none seem to work. I've also tried tweeking each one with my limited knowledge of applescript to no avail. It really cannot be that hard for the computer to say "hey, there's a new message in that folder, I'll save its PDF to my desktop." Can it?

This is the basics of my goal. Ideally, the applescript/rule/computer would add the Received Date to the attachment name and would know whether it had already saved a given email's attachment(s), so as to reduce overwriting, possibly through read-status or flagging.