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The role of compressible fluctuations in Hall magnetohydrodynamics (HMHD) turbulence is investigated using in-situ measurements in the terrestrial magnetosheath and direct numerical simulations. A focus is put on verifying the recent exact law derived for compressible isothermal HMHD turbulence (Andres, Galtier and Sahraoui, 2018). Observations from the recently launched multispacecraft NASA/MMS (Magnetospheric MultiScale) mission in the solar wind are analized. In particular, to compute the different terms in the exact law multispacecraft techniques are used. Direct comparison with numerical simulations (3D compressible HMHD code) are made. The main goal is to estimate the energy cascade rate of turbulence over a broad range of scales that span both the inertial and sub-ion (dispersive) ranges. Moreover, the energy cascade rate is strongly affected by the presence of density fluctuation and, also, it have been found that the role of the compressible flux term has a huge impact in the nonlinear dynamics as it is increased the ion density fluctuations.