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In addition to the Kitaev ($K$) interaction, candidate Kitaev materials alsopossess Heisenberg ($J$) and off-diagonal symmetric ($\Gamma$) couplings. Weinvestigate the quantum ($S = 1/2$) $K$-$J$-$\Gamma$ model on the honeycomblattice by a variational Monte Carlo (VMC) method. In addition to the "generic"Kitaev spin liquid (KSL), we find that there is just one proximate KSL (PKSL)phase, while the rest of the phase diagram contains different magneticallyordered states. The PKSL is a gapless Z$_2$ state with 14 Majorana cones, whichin contrast to the KSL has a gapless spin response. In a magnetic field appliednormal to the honeycomb plane, it realizes two of Kitaev's gapped chiralspin-liquid phases, of which one is non-Abelian with Chern number $\nu = 5$ andthe other is Abelian with $\nu = 4$. These two phases could be distinguished bytheir thermal Hall conductance.

One Proximate Kitaev Spin Liquid in the $K$-$J$-$\Gamma$ Model on the Honeycomb Lattice