  In [1] we proved a strengthened Kochen-Specker theorem in 3 dimensions:
non-contextual hidden variable (NCHV) models cannot reproduce all the quantum
correlations of two compatible observables, which is a minimal requirement
imposed on the NCHV models. Here we shall exclude the NCHV models with this
minimal requirement in $d\ge 4$ dimensions by state-independent and
experimentally testable inequalities satisfied by all NCHV models that are
required to reproduce only the quantum correlations of at most two compatible
observables. Furthermore our proofs use the smallest number of rays known so
far, e.g., 25 (instead of 31) rays in $5$ dimensions and $5d-2\lfloor
d/3\rfloor$ rays in $d\ge 6$ dimensions.
