  The particle production at very forward rapidities is expected to be
sensitive to the non-linear effects in the QCD dynamics at high energies. In
this paper we present, for the first time, the predictions of the Color Dipole
formalism for the single inclusive jet photoproduction in $pp$ and $pPb$
collisions considering the very forward rapidities probed by the CMS-CASTOR
calorimeter, which will be characterized by a jet in the rapidity range of $5.2
\le Y \le 6.6$, a rapidity gap in the rapidity range probed by the central CMS
detector and one of the incident hadrons remaining intact in the final state.
The transverse momentum distributions are estimated considering the more recent
phenomenological models for the dipole-proton scattering amplitude, which are
based on the Color Glass Condensate formalism and are able to describe the
inclusive and exclusive $ep$ HERA data. Our results indicate that a future
experimental analysis of this process is, in principle, feasible and useful to
constrain the description of the QCD dynamics at high energies.
