The centre of the Milky Way hosts the most massive and dense clouds of molecular hydrogen gas in our Galaxy. The inferred star formation efficiency at the Galactic Centre is however surprisingly low, given the large gas reservoir. Uncertainties in the measurement of gas density makes the comparison between observations and theories difficult. Measurements of the gas density based on different mass tracers yield inconsistent results, even when using conventional probes such as CS and dust. We propose using neutrinos as an alternative, independent gas tracer to resolve this ambiguity. Neutrinos are produced when cosmic rays interact with the cloud gas, with their surface brightness directly proportional to the cloud's density integrated along the line of sight. We quantify the impact of future neutrino data from the next generation of telescopes—KM3NeT, Baikal-GVD, and P-ONE—which offer angular resolutions close to a tenth of a degree for detecting muon neutrinos from the Galactic Centre. We show how neutrinos will improve the measurement of the gas density, allowing for more robust tests of star formation theories.