  We prove the Kuniba-Nakanishi-Suzuki (KNS) conjecture concerning the quantum
dimension solution of the $Q$-system of type $D_r$ obtained by a certain
specialization of classical characters of the Kirillov-Reshetikhin modules. To
this end, we use various symmetries of quantum dimensions. As a result, we
obtain an explicit formula for the positive solution of the level $k$
restricted $Q$-system of type $D_r$ which plays an important role in
dilogarithm identities for conformal field theories.
