  For an even qudit dimension $d\geq 2,$ we introduce a class of two-qudit
states exhibiting perfect correlations/anticorrelations and prove via the
generalized Gell-Mann representation that, for each two-qudit state from this
class, the maximal violation of the original Bell inequality is bounded from
above by the value $3/2$ - the upper bound attained on some two-qubit states.
We show that the two-qudit Greenberger-Horne-Zeilinger (GHZ) state with an
arbitrary even $d\geq 2$ exhibits perfect correlations/anticorrelations and
belongs to the introduced two-qudit state class. These new results are
important steps towards proving in general the $\frac{3}{2}$ upper bound on
quantum violation of the original Bell inequality. The latter would imply that
similarly as the Tsirelson upper bound $2\sqrt{2}$ specifies the quantum analog
of the CHSH inequality for all bipartite quantum states, the upper bound
$\frac{3}{2}$ specifies the quantum analog of the original Bell inequality for
all bipartite quantum states with perfect correlations/ anticorrelations.
Possible consequences for the experimental tests on violation of the original
Bell inequality are briefly discussed.
