  Recently, V\'{e}rtesi and Bene [Phys. Rev. A. {\bf 82}, 062115 (2010)]
derived a two-qubit Bell inequality, $I_{CH3}$, which they show to be maximally
violated only when more general positive operator valued measures (POVMs) are
used instead of the usual von Neumann measurements. Here we consider a general
parametrization for the three-element-POVM involved in the Bell test and obtain
a higher quantum bound for the $I_{CH3}$-inequality. With a higher quantum
bound for $I_{CH3}$, we investigate if there is an experimental setup that can
be used for observing that POVMs give higher violations in Bell tests based on
this inequality. We analyze the maximum errors supported by the inequality to
identify a source of entangled photons that can be used for the test. Then, we
study if POVMs are also relevant in the more realistic case that partially
entangled states are used in the experiment. Finally, we investigate which are
the required efficiencies of the $I_{CH3}$-inequality, and the type of
measurements involved, for closing the detection loophole. We obtain that POVMs
allow for the lowest threshold detection efficiency, and that it is comparable
to the minimal (in the case of two-qubits) required detection efficiency of the
Clauser-Horne-Bell-inequality.
