An analysis of the breakup of the 31Ne weakly-bound neutron-halo system on a lead target is presented, considering 2p3/2 and 1f7/2 ground-state configurations. It is shown that a high centrifugal barrier almost wipes out the breakup channel, thus assimilating the breakup of a weakly-bound system to that of a tightly-bound system, and also reduces the range of the monopole nuclear potential. Consequently, a high centrifugal barrier prevents the suppression of the Coulomb-nuclear interference (CNI) peak by weakening the breakup channel, and by reducing the range of the monopole nuclear potential, two main factors that would otherwise suppress that peak. In conclusion, the present study also identifies couplings to the breakup channel and a long-ranged monopole nuclear potential as the main factors that lead to the suppression of the CNI peak. A low centrifugal barrier together with a Coulomb barrier would as well effectively prevent the suppression of the CNI peak in proton-halos, as it has been reported the case of the 8B proton-halo.