After extensive troubleshooting, google searching, and HP support chats, there didn't seem to be any improvement, and the issue still seemed to be hardware based. Just before resigning myself to sending the machine to HP for repair, I found a suggestion buried in a large thread on a message board, which I decided to try out. After a couple of days, I'm happy to report that it's working beautifully!
As it turns out, theres a wire under the keyboard that needs to be in contact with the ground plate, and by default is not actually fastened to it in any way, but depends on pressure from the components above. If the laptop is shifted in such a way that the wire loses contact with that plate for a moment, the problem triggers. Poor design? Yes. The good news is that it's easily fixed! Here's how:
Firstly, follow the instructions in this document to remove the keyboard. At that point you should be looking at something like this:
The red box indicates the wire mentioned above. A better view is here:
All that is needed is to secure the foil above to the plate just below it. I used masking tape, and everything seems to be working perfectly.
A word of warning though: when removing the speaker plate, do not try to pry the silver side plates of the machine away to facilitate the process, as the clasps that hold it in place are very delicate. I have a broken piece of plastic bouncing off of my cooling fan now and then, and its a bit disconcerting. Worth it though, for a working touch panel.
Good luck on your efforts, and comment below if you find this helpful :-)