I was reading this paper "Quantum physics needs complex numbers" by Renou et al. In the paper, they propose the standard entanglement swapping scenario and use CHSH3 inequality to show that the maximal bound predicted by quantum mechanics can't be achieved. I have two questions regarding this
Why did the authors use the CHSH3 inequality instead of the CHSH inequality. Real quantum mechanics should also not be achieve maximal violation of CHSH so why CHSH3?
Also is there a more intuitive explanation for why the reall simulation fails in the entanglement swapping scenario but works fine in the Bell experiment scenario? The authors do provide a reason but I didn't quite find it so intuitive)