Assuming the definition of the inversion problem (IP) as the exact matching of the terms in the low redshift expansion of cosmological observables calculated for different cosmological models, we solve the IP for ${D}_{L}(z)$ and the redshift spherical shell mass density $mn(z)$ for a central observer in a Lemaitre-Tolman-Bondi (LTB) space without cosmological constant and a generic $\ensuremath{\Lambda}\mathrm{CDM}$ model. We show that the solution of the IP is unique, corresponds to a matter density profile which is not smooth at the center, and that the same conclusions can be reached expanding self-consistently to any order all the relevant quantities. Contrary to the case of a single observable inversion problem, it is impossible to solve the IP (LTB vs $\ensuremath{\Lambda}\mathrm{CDM}$) for both $mn(z)$ and ${D}_{L}(z)$ while setting one of the two functions $k(r)$ or ${t}^{b}(r)$ to zero, even allowing nonsmooth matter profiles. Our conclusions are general, since they are exclusively based on comparing directly physical observables in redshift space, and do not depend on any special ansatz or restriction for the functions defining a LTB model.