A Conjecture on the Amount of Non-Locality

Imagine a world in which there exist physical resources for non-local correlations whose CHSH value lies between 2 and $X\leq 4$. Assume that such resources can be mixed in some sense. Using Connes's result on the extension of characteristic 1 semi-rings, we conjecture a possible motivation for quantum mechanical resources obeying the Tsirelson bound $X=2\sqrt{2}$.