  The Sachdev--Ye--Kitaev is a quantum mechanical model of $N$ Majorana
fermions which displays a number of appealing features -- solvability in the
strong coupling regime, near-conformal invariance and maximal chaos -- which
make it a suitable model for black holes in the context of the AdS/CFT
holography. In this paper, we show for the colored SYK model and several of its
tensor model cousins that the next-to-leading order in the large $N$ expansion
preserves the conformal invariance of the $2$-point function in the strong
coupling regime, up to the contribution of the pseudo-Goldstone bosons due to
the explicit breaking of the symmetry and which are already seen in the leading
order $4$-point function. We also comment on the composite field approach for
computing correlation functions in colored tensor models.
