  We consider a dilute gas of hard spheres under shear. We use one of the
predominant models to study this system, namely the SLLOD equations of motion,
with an iso-kinetic Gaussian thermostat in between collisions, to get a
stationary total peculiar kinetic energy. Based on the previously obtained
result that in the non-equilibrium steady state and in case the number of
particles $N$ becomes large, the coefficient of dynamical friction representing
the iso-kinetic Gaussian thermostat for the SLLOD dynamics fluctuates with
$1/\sqrt{N}$ fluctuations around a fixed value, we show on analytical grounds
that for a hard sphere gas at small shear rate and with a large number of
spheres, the conjugate pairing of the Lyapunov exponents is expected to be
violated at the fourth power of the constant shear rate in the bulk.
