The headphone jack is analog, HDMI is digital. That would NEVER be able to happen,a nd even in the RARE (stupid) occasion someone took the time to code something that COULD push HDMI signals THROUGH analog you damn for sure wouldn't get HD. You'd get 480p at best.
It would be a waste of time for the coder, and poor sap that use it IMHO. You'd have better luck pushing HDMI signals through the USB port.
Albeit the usb port is a charging port but it's still a media port at heart. There's more a chance of you being able to push some sort of composite, or digital signal through it than you could through the headphones.