We numerically demonstrate that, although it is critical, the two-box symmetric SU(3) chain possesses edge states in the adjoint representation whose excitation energy scales with the number of sites Ns as 1/(Ns log Ns), in close analogy to those found in half-integer SU(2) chains with spin S 3/2. We further show that these edge states dominate the entanglement entropy of finite chains, explaining why it has been impossible so far to verify with density-matrix renormalization group simulations the field theory prediction that this model is in the SU(3)1 universality class. Finally, we show that these edge states are very efficiently screened by attaching adjoint representations at the ends of the chain, leading to an estimate of the central charge consistent within 1% with the prediction c = 2 for SU(3)1.