  We study the response of a two-channel Kondo quantum dot device proposed by
Y. Oreg and D. Goldhaber-Gordon [Phys. Rev. Lett. {\bf 90}, 136602 (2003)] to a
spin-bias applied across one of its channels formed by Fermi liquid reservoirs
weakly coupled to a spin-1/2 quantum dot. When the temperature $T<T_K$, the
Kondo temperature of the device, the spin conductance depends on the Kondo
coupling of the dot spin to the other channel in an anomalous manner. For
isotropic Kondo couplings to the two channels the spin conductance is quantized
for $T\ll T_K$ characterizing the two-channel Kondo fixed point. On the other
hand, for anisotropic couplings a crossover energy scale $T_A\neq 0$ determines
the temperature $T\ll T_A$ when the spin conductance vanishes indicating
one-channel Kondo behavior.
