  We study the nonequilibrium time evolution of the spin-1/2 anisotropic
Heisenberg (XXZ) spin chain, with a choice of dimer product and Neel states as
initial states. We investigate numerically various short-ranged spin
correlators in the long-time limit and find that they deviate significantly
from predictions based on the generalized Gibbs ensemble (GGE) hypotheses. By
computing the asymptotic spin correlators within the recently proposed
quench-action formalism [Phys. Rev. Lett. 110, 257203 (2013)], however, we find
excellent agreement with the numerical data. We, therefore, conclude that the
GGE cannot give a complete description even of local observables, while the
quench-action formalism correctly captures the steady state in this case.
