A carbon nanofiber material, consisting of bottomless graphene cups inside on each other in a line, like a set of soft-drink cups, has been discovered to have the potential to conduct heat ballistically over a long distance. Its longitudinal heat transport ability had been forecast to be extremely poor due to the weak van der Waals force operating between the graphene cups, but our measurements using nano thermal sensor showed that its thermal conductivity is much higher than that along the c-axis of bulk graphite. This unexpected result can be understood by its similarity to a one-dimensional (1D) harmonic-chain where no phonon is scattered even for an infinite length. The current graphene-based nanofiber resembles this type of "superconductive" chain due to the huge difference between the stiff covalent bonding in each cup and the weak inter-cup interaction. A non-equilibrium molecular dynamics simulation is conducted to explore the phonon transport in this fiber. The simulation results show that the thermal conductivity varies with the fiber length in a power law fashion with an exponent as large as 0.7. The calculated phonon density of states and atomic motions indicate that a low-frequency quasi-1D oscillation occurs there. Our investigations show that treating the current nanofiber as a 1D chain with three-dimensional oscillations explains well why this material has the most effective ballistic phonon transport ever observed.