On irreversibility of von Neumann additive cellular automata on grids

The von Neumann cellular automaton appears in many different settings in Operations Research varying from applications in Formal Languages to Biology. One of the major questions related to it is to find a general condition for irreversibility of a class of two-dimensional cellular automata on square grids (σ+σ+-automata  ). This question is partially answered here with the proposal of a sufficient condition for the irreversibility of σ+σ+-automata.