I've read your first post very very well. I did all the research about your old and your new projector. I did the research of HDMI cables from 2007 (nine years ago). I provided pretty much all you need to know, for today and tomorrow.
The rest is up to you; me I just give you my best time. Never I told you what to do, I simply shared info...the best info to my knowledge from my research.
* Caelin's last line from above: "It sounds like replacing the cable in the wall would be a royal pain. So, if it is a 1.3 cable it is very likely that it will work at full resolution assuming that the length is not excessive. Give it a try."
If it works fine, if it doesn't you have all the info now.
Here's something else: I've read comments from hi-end videophiles who had great picture quality until they replace their HDMI cable with superior ones and bang, the picture quality, and the sound quality too, improved. But that, is easy to test; you don't have to open your walls and ceiling for that.
* Do you have the exact model/brand of your HDMI cable right now, and how many feet long is it? What HDMI source(s) are you using?
I guess not, as you said on your first post.
HDMI cables are not all created equal, and there is too much confusion still today about compatibility between them and the audio/video components used.
I can see few years from now...with 8K the day high-end HDMI cables will sell for thousands of dollars.