Can the degeneracies in the gravity sector be broken?

It has been long known that by appropriately modifying gravity one can always reproduce the expansion history of any dark energy model, e.g. the $\Lambda$CDM. This degeneracy cannot be broken with geometric probes like the Type Ia supernovae, the cosmic microwave background or the baryon acoustic oscillations since they are based on the measurement of distances and scales and therefore require the expansion history $H(z)$, so one may only hope to break this degeneracy by using dynamic probes like the growth rate data that track the matter density perturbations. We demonstrate that breaking this degeneracy is not currently possible by explicitly constructing $f(R)$, $f(G)$ and $f(T)$ theories that mimic exactly the $\Lambda$CDM model at the background level and confronting them against the latest observational cosmological data. We also determine the necessary improvement in the growth rate data in order to discriminate these theories from $\Lambda$CDM. We found that at least a threefold improvement is necessary, something which will be possible with a survey like Euclid.