  We revisit the SU(6) Heisenberg model on the honeycomb lattice, which has
been predicted to be a chiral spin liquid by mean-field theory [G. Szirmai et
al., Phys. Rev. A 84, 011611 (2011)]. Using exact diagonalizations of finite
clusters, infinite projected entangled pair states simulations, and variational
Monte Carlo simulations based on Gutzwiller projected wave functions, we
provide strong evidence in favour of the competing plaquette state, which was
reported to be higher but close by in energy according to mean-field theory.
This is further confirmed by the investigation of the model with a ring
exchange term, which shows that there is a transition between the plaquette
state and the chiral state at a finite value of the ring exchange term.
