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© 2016, The Author(s).We study the general structure of correlation functions in an Sp(2n)-invariant formulation of systems of an infinite number of higher-spin fields. For n = 4, 8 and 16 these systems comprise the conformal higher-spin fields in space-time dimensions D = 4, 6 and 10, respectively, while when n = 2, one deals with conventional D = 3 conformal field theories of scalars and spinors. We show that for n > 2 the Sp(2n) symmetry and current conservation makes the 3-point correlators of two (rank-one or rank-two) conserved currents with a scalar operator be that of free theory. This situation is analogous to the one in conventional conformal field theories, where conservation of higher-spin currents implies that the theories are free.