We consider a minimal interacting theory of a single tower of spin j = 0, 2, 4,… massless Fronsdal fields in flat space with local Lorentz-covariant cubic interaction vertices. We address the question of constraints on possible quartic interaction vertices imposed by the condition of on-shell gauge invariance of the tree-level four-point scattering amplitudes involving three spin 0 and one spin j particle. We find that these constraints admit a local solution for quartic 000j interaction term in the action only for j = 2 and j = 4. We determine the non-local terms in four-vertices required in the j ≥ 6 case and suggest that these non-localities may be interpreted as a result of integrating out a tower of auxiliary ghost-like massless higher spin fields in an extended theory with a local action, up to possible higher-point interactions of the ghost fields. We also consider the conformal off-shell extension of the Einstein theory and show that the perturbative expansion of its action is the same as that of the non-local action resulting from integrating out the trace of the graviton field from the Einstein action. Motivated by this example, we conjecture the existence of a similar conformal off-shell extension of a massless higher spin theory that may be related to the above extended theory. It may then have the same infinite-dimensional symmetry as the higher-derivative conformal higher spin theory and may thus lead to a trivial S matrix.