Using the Karpman-Solov'ev method we derive the equations for the 2-soliton adiabatic interaction for solitons of the modified nonlinear Schrödin-ger equation (MNSE). Then we generalize these equations to the case of N interacting solitons with almost equal velocities and widths. On the basis of this result we prove that the N MNSE-soliton train interaction (N > 2) can be modeled by the completely integrable complex Toda chain (CTC). This is an argument in favor of universality of the complex Toda chain which was previously shown to model the soliton train interaction for nonlinear Schrödinger solitons. The integrability of the CTC is used to describe all possible dynamical regimes of the N -soliton trains which include asymptotically free propagation of all N solitons, N -soliton bound states, various mixed regimes, etc. It allows also to describe analytically the manifolds in the 4N -dimensional space of initial soliton parameters which are responsible for each of the regimes mentioned above. We compare the results of the CTC model with the numerical solutions of the MNSE for 2 and 3-soliton interactions and find a very good agreement.