Forward $J/\psi$ and very backward jet inclusive production at the LHC

In the spirit of Mueller-Navelet dijet production, we propose and study the inclusive production of a forward $J/\psi$ and a very backward jet at the LHC as an observable to reveal high-energy resummation effects \`a la BFKL. We obtain several predictions, which are based on the various mechanisms discussed in the literature to describe the production of the $J/\psi$, namely, NRQCD singlet and octet contributions, and the color evaporation model.