Which 2P1R Games are Potentially Sharp?
I tend to believe that c=3 is the right answer for the general case, and that it should be possible to give an example. I'll have to think more about that to know for sure.
It's a good question, and I don't know of existing work about it. 

Research recently focused on which types of games have (best possible) c=1, mostly because of possible applications to amplification of unique games.


Barak et al generalized the counterexample of Raz to all unique games with SDP gaps.
Raz and Rosen showed that for expanding projection games c=1. There are also previous results by a super-set of those authors for free games.