  We consider the conformal higher spin (CHS) theory in d=4 that contains the
s=1 Maxwell vector, s=2 Weyl graviton and their higher spin s=3,4,...
counterparts with higher-derivative \box^s kinetic terms. The interacting
action for such theory can be found as the coefficient of the logarithmically
divergent part in the induced action for sources coupled to higher spin
currents in a free complex scalar field model. We explicitly determine some
cubic and quartic interaction vertices in the CHS action from scalar loop
integrals. We then compute the simplest tree-level 4-particle scattering
amplitudes 11 -> 11, 22 -> 22 and 11 -> 22 and find that after summing up all
the intermediate CHS exchanges they vanish. This generalises the vanishing of
the scattering amplitude for external conformal scalars interacting via the
exchange of all CHS fields found earlier in arXiv:1512.08896. This vanishing
should generalise to all scattering amplitudes in the CHS theory and as in the
conformal scalar scattering case should be a consequence of the underlying
infinite dimensional higher spin symmetry that extends the standard conformal
symmetry.
