It is a well-established fact that any conformal field theory with a gap in the twist spectrum must contain families of multi-twist operators, whose spectrum at large spin approaches that of generalized free theory. In this talk, we aim to discuss how the lightcone bootstrap can be applied to five- and six-point correlation functions in the comb channel to constrain the behavior of double- and triple-twist operators. Our analysis yields new expressions for large-spin OPE coefficients involving two double-twist operators, as well as the leading-order anomalous dimension matrix for triple-twist operators. The latter offers valuable insight into how the degeneracy of triple-twist primaries in the free-theory limit gets lifted by the inclusion of interactions.