  The influence of the electron-vibron coupling on the transport properties of
a strongly interacting quantum dot built in a suspended carbon nanotube is
analyzed. The latter is probed by a charged AFM tip scanned along the axis of
the CNT which induces oscillations of the chemical potential and of the linear
conductance. These oscillations are due to the competition between finite-size
effects and the formation of a Wigner molecule for strong interactions. Such
oscillations are shown to be suppressed by the electron-vibron coupling. The
suppression is more pronounced in the regime of weak Coulomb interactions,
which ensures that probing Wigner correlations in such a system is in principle
possible.
