  We analyze the first measurement of $\eta_c$ production, performed by the
LHCb Collaboration, in the nonrelativistic-QCD (NRQCD) factorization framework
at next-to-leading order (NLO) in the strong-coupling constant $\alpha_s$ and
the relative velocity $v$ of the bound quarks including the feeddown from $h_c$
mesons. Converting the long-distance matrix elements (LDMEs) extracted by
various groups from $J/\psi$ yield and polarization data to the $\eta_c$ case
using heavy-quark spin symmetry, we find that the resulting NLO NRQCD
predictions greatly overshoot the LHCb data, while the color-singlet model
provides an excellent description.
