I've seen this issue documented time and time again, but each time seems to result in a new fix (the newest being to run Game Capture HD 2.1). My setup is as follows:

PS4 into Elgato Game Capture HD (not the 60) Mac Pro 2012: 2 x 2.66 GHz 6-Core Intel Xeon, 16GB RAM, and while I'm mostly streaming and not doing any local capture, I have 2 Samsung 840 Pro SSDs in RAID 0 to write to.

I'm currently running 2.1b85, although the issue seems to exist on every version of the software imaginable. I get about 1800ms of delay that I adjust for, but as my stream goes on the delay becomes 3 seconds, 4 seconds, etc. It's gotten to the point where if I don't restart my stream about every 45 minutes or so then the PS4 audio is completely out of sync. I'm only streaming at 720p, I'm not capturing anything locally, and I've made sure the Flashback Recording option is off. I'm just capturing the Game Capture HD software Window using GameShow (I was using OBS previously, same issue). Any help or insight would be greatly appreciated!