Before revision:  Finally  we give a polynomial-time reduction from c-planarity to thickenability and show that a slight generalization of atomic embeddability to the setting in which clusters are toroidal graphs is NP-complete.
After revision:  Finally we give a polynomial-time reduction from atomic embeddability to thickenability thereby showing that both problems are polynomially equivalent, and show that a slight generalization of atomic embeddability to the setting in which clusters are toroidal graphs is NP-complete.