Reproducing the large mass ratio between the Earth and Mars requires that the terrestrial planets formed from a narrow annulus, with a steep mass density gradient beyond 1 AU (Hansen, 2009). The Grand Tack scenario (Walsh et al., 2011) invokes a specific migration history of the giant planets of the Solar System to remove most of the mass initially beyond 1 AU and to leave the asteroid belt on an excited dynamical state. However, one could also invoke that the steep mass density gradient was achieved by the migration and pile-up of a large amount of small particles induced by gas-drag. This process has been proposed to explain the formation of close-in super Earths in extrasolar systems (e.g. Chatterjee and Tan, 2015). Here we show that the asteroid belt orbital excitation provides a crucial constraint against this scenario for the Solar System. We achieve this result by performing a series of numerical simulations of terrestrial planet formation and asteroid belt evolution, starting from disks of planetesimals and planetary embryos with various radial density gradients. Jupiter and Saturn are assumed on their current, non-migrating orbits. We find that disks with shallow density gradients allow the dynamical excitation of the asteroid belt by a self-stirring process, but lead inevitably to the formation of a Mars analog which is significantly more massive than the real planet. Instead, a disk with a surface density gradient proportional to 1/r^5 beyond 1 AU allows us to reproduce the Earth/Mars mass ratio, but leaves the asteroid belt on a dynamical state way too cold compared to the real belt. Therefore, we conclude that no disk profile can explain at the same time the structure of the terrestrial planet system and of the asteroid belt. Thus, the asteroid belt has to have been depleted and dynamically excited by an external agent as, for instance, in the Grand Tack scenario.