For both DMABN and fulvene, we demonstrated that CCTTSH cures the internal consistency issue of CTTSH and yields a stable nuclear dynamics, for instance in terms of nuclear kinetic energy. These observations, especially on the challenging DMABN, together with the robustness of the results by varying the scheme for velocity rescaling reported in Ref. [39], confirm that CCTTSH is a promising method for NAMD. DMABN and fulvene highlighted some issues in CTTSH, probably inherited from TSH, that were only observed previously in a twodimensional model for phenol , but apparently persist in higher dimensions.