Mars is thought to have hosted large amounts of water and carbon dioxide at primitive epochs. The morphological analysis of the surface of Mars shows that large bodies of water were probably present in the North hemisphere at late Noachian (3.7–4 Gyr ago). Was this water solid or liquid? For maintaining liquid water at this time, when the Sun was (likely) less bright than now, a CO2 atmosphere of typically 2 bars is required. Can sputtering, still presently acting at the top of the Martian atmosphere, have removed such a dense atmosphere over the last 3.5–4 Gyr? What was the fate of the 100–200 m global equivalent layer of water present at late Noachian? When did Martian magnetic dynamo vanish, initiating a long period of intense escape by sputtering? Because sputtering efficiency is highly non-linear with solar EUV flux, with a logarithmic slope of ≈7:Φsput≈ΦEUV7, resulting in enhanced levels of escape at primitive epochs, when the sun was several times more luminous than now in the EUV, there is a large uncertainty on the cumulated amount of volatiles removed to space. This amount depends primarily on two factors: (i) the exact value of the non-linearity exponent (≈7 from existing models, but this value is rather uncertain), (ii) the exact time when the dynamo collapsed, activating sputtering at epochs when intense EUV flux and solar wind activity prevailed in the solar system. Both parameters are only crudely known at the present time, due the lack of direct observation of sputtering from Martian orbit, and to the incomplete and insufficiently spatially resolved map of the crustal magnetic field. Precise timing of the past Martian dynamo can be investigated through the demagnetisation signature associated with impact craters. A designated mission to Mars would help in answering this crucial question: was water liquid at the surface of Mars at late Noachian? Such a mission would consist of a low periapsis (≈100 km) orbiter, equipped with a boom-mounted magnetometer, for mapping the magnetic field, as well as adequate in situ mass and energy spectrometers, for a full characterization of escape and of its response to solar activity variations. Surface based observations of atmospheric noble gas isotopic ratios, which keep the signatures of past escape processes, including sputtering for the lightest of them (Ne, Ar), would bring a key constraint for escape models extrapolated back to the past.