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The dynamical structure of the Kuiper belt can be used as a clue to the formation and evolution of the Solar System, planetary systems in general, and Neptune's early orbital history in particular. The problem is best addressed by forward modeling where different initial conditions and Neptune's orbital evolutions are tested, and the model predictions are compared to orbits of known Kuiper belt objects (KBOs). It has previously been established that Neptune radially migrated, by gravitationally interacting with an outer disk of planetesimals, from the original radial distance r < 25 au to its current orbit at 30 au. Here we show that the migration models with a very low orbital eccentricity of Neptune e_N < 0.05 do not explain KBOs with semimajor axes 50 < a < 60 au, perihelion distances q > 35 au and inclinations i < 10 deg. If e_N < 0.05 at all times, the Kozai cycles control the implantation process and the orbits with q > 35 au end up having, due to the angular momentum's z-component conservation, i > 10 deg. Better results are obtained when Neptune's eccentricity is excited to e_N ~ 0.1 and subsequently damped by dynamical friction. The low-e & low-i orbits at 50-60 au are produced in this model when KBOs are lifted from the scattered disk by secular cycles -- mainly the apsidal resonance nu_8 -- near various mean motion resonances. These results give support to a (mild) dynamical instability that presumably excited the orbits of giant planets during Neptune's early migration.