  Hardy-type arguments manifest Bell nonlocality in one of the simplest
possible ways. Except for demonstrating nonclassical signature of entangled
states in question, they can also serve for device-independent self-testing of
states, as shown, e.g., in Phys. Rev. Lett. 109, 180401 (2012). Here we develop
and broaden these results to an extended version of Hardy's argument, often
referred to as Cabello's nonlocality argument. We show that, as in the simpler
case of Hardy's nonlocality argument, the maximum quantum value for Cabello's
nonlocality is achieved by a pure two-qubit state and projective measurements
that are unique up to local isometries. We also examine the properties of a
more realistic case when small errors in the ideal constraints are accepted
within the probabilities obtained and prove that also in this case the
two-qubit state and measurements are sufficient for obtaining the maximum
quantum violation of the classical bound.
