  The degree of experimentally attainable nonlocality, as gauged by the amount
of loophole-free violation of Bell inequalities, remains severely limited due
to inefficient detectors. We address an experimentally motivated question:
Which quantum strategies attain the maximal loophole-free nonlocality in the
presence of inefficient detectors? For any Bell inequality and any
specification of detection efficiencies, the optimal strategies are those that
maximally violate a tilted version of the Bell inequality in ideal conditions.
In the simplest scenario, we demonstrate that the quantum strategies that
maximally violate the tilted versions of Clauser-Horne-Shimony-Holt inequality
are unique up to local isometries. However, self-testing via the standard sum
of squares decomposition method turns out to be analytically intractable since
even high levels of the Navascu\'es--Pironio--Ac\'in hierarchy are insufficient
to saturate the maximum quantum violation of these inequalities. Instead, we
utilize a novel Jordan's lemma-based proof technique to obtain robust
analytical self-testing statements for the entire family of tilted-Bell
inequalities. These results allow us to unveil intriguing aspects of the effect
of inefficient detectors and the complexity of characterizing the set of
quantum correlations, in the simplest Bell scenario.
