  Quantum correlations are contextual yet, in general, nothing prevents the
existence of even more contextual correlations. We identify and test a
noncontextuality inequality in which the quantum violation cannot be improved
by any hypothetical postquantum theory, and use it to experimentally obtain
correlations in which the fraction of noncontextual correlations is less than
0.06. Our correlations are experimentally generated from the results of
sequential compatible tests on a four-state quantum system encoded in the
polarization and path of a single photon.
