  In the free $\Box^k$ scalar conformal field theory, there exist conserved and
partially-conserved higher-spin currents. We study their anomalous dimensions
associated with $\phi^{2n}$ interaction in the $\epsilon$ expansion. We derive
general formulae for the leading corrections from the conformal multiplet
recombination, and verify their consistency with crossing symmetry using the
Lorentzian inversion formula. The results are further extended to the O($N$)
models.
