  In this review we discuss and extend the study of the inclusive production of
vector quarkonia, $J/\psi$ and $\Upsilon$, emitted with large transverse
momenta and rapidities at the LHC. We adopt the novel ZCW19$^+$ determination
to depict the quarkonium production mechanism at the next-to-leading level of
perturbative QCD. This approach is based on the nonrelativistic QCD formalism
well adapted to describe the production of a quarkonium state from the
collinear fragmentation of a gluon or a constituent heavy quark at the lowest
energy scale. We rely upon the NLL/NLO$^+$ hybrid high-energy and collinear
factorization for differential cross sections, where the standard collinear
formalism is enhanced by the BFKL resummation of next-to-leading energy
logarithms arising in the $t$-channel. We employ the JETHAD method to analyze
the behavior of rapidity distributions for double inclusive vector-quarkonium
and inclusive vector-quarkonium plus jet emissions. We discovered that the
natural stability of the high-energy series, previously observed in observables
sensitive to the emission of hadrons with heavy flavor detected in the rapidity
acceptance of LHC barrel calorimeters, becomes even more manifest when these
particles are tagged in forward regions covered by endcaps. Our findings brace
the important message that vector quarkonia at the LHC via the hybrid
factorization offer a unique chance to perform precision studies of high-energy
QCD, as well as an intriguing opportunity to shed light on the quarkonium
production puzzle.
