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We study the behaviour of weak gravitational fields in the 6D Cascading DGPmodel using a bulk-based approach. To deal with the ambiguity in the thin limitof branes of codimension higher than one, we consider a specific regularizationof the internal structure of the branes where the 5D brane can be consideredthin with respect to the 4D one. We consider the solutions corresponding topure tension sources on the 4D brane, and study perturbations at first orderaround these background solutions. We adopt a 4D scalar-vector-tensordecomposition, and focus on the scalar sector of perturbations. We show that,in a suitable 4D limit, the trace part of the 4D metric perturbations obeys adecoupled equation which suggests that it is a ghost for background tensionssmaller than a critical tension, while it is a healthy field otherwise. We givea geometrical interpretation of the existence of the critical tension and ofthe reason why the relevant field is a ghost or not depending on the backgroundtension. We however find a value of the critical tension which is differentfrom the one already found in the literature. Differently from the results inthe literature, our analysis implies that, choosing the background tensionsuitably, we can construct ghost-free models for any value of the freeparameters of the theory. We suggest that the difference lies in the procedureused to evaluate the pillbox integration across the codimension-2 brane. Weconfirm the validity of our analysis by performing numerically the integrationin a particular case where the solution inside the thick cod-2 brane is knownexactly. We stress that the singular structure of the perturbation fields inthe nested branes set-ups is very subtle, and that great care has to be takenwhen deriving the codimension-2 junction conditions.

The critical tension in the Cascading DGP model