  Motivated by the possibility of finding a bosonic quantum spin liquid in the
integer spin-$S$ Heisenberg-Kitaev model on the honeycomb lattice, we derive a
Schwinger boson mean field theory involving both singlet and triplet pairing
channels which includes hopping and pairing operators on equal footing. The
mixed construction introduced here is justified by the good comparison with
exact diagonalization energies of the $S \leq 3/2$ Heisenberg-Kitaev model and
the perfect match with the Luttinger-Tisza semiclassical energies obtained at
large-$S$. We find various competing gapped quantum spin liquids close to the
Kitaev point. A comparison of their spin excitation spectrum with the dynamical
structure factor obtained from exact diagonalizations allows us to identify the
physical spin liquid $ans\"atz$ of the model. In particular, we identify a
chiral quantum spin liquid state whose spin excitation spectrum follows closely
the exact diagonalization data and survives up to large spin $S \lesssim 2$. We
propose this state as a promising quantum spin liquid candidate for the integer
spin-$S$ antiferromagnetic Kitaev model which may be realized in $S=1$ Kitaev
materials A$_3$Ni$_2$XO$_6$ and KNiAsO$_4$.
