  We present a much simplified version of the CGLMP inequality for the 2 x 2 x
d Bell scenario. Numerical maximization of the violation of this inequality
over all states and measurements suggests that the optimal state is far from
maximally entangled, while the best measurements are the same as conjectured
best measurements for the maximally entangled state. For very large values of d
the inequality seems to reach its minimal value given by the probability
constraints. This gives numerical evidence for a tight quantum Bell inequality
(or generalized Csirelson inequality) for the 2 x 2 x inf scenario.
