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Seven temperate Earth-sized exoplanets readily amenable for atmospheric studies transit the nearby ultracool dwarf star TRAPPIST-1 (refs ^,). Their atmospheric regime is unknown and could range from extended primordial hydrogen-dominated to depleted atmospheres^{, , –}. Hydrogen in particular is a powerful greenhouse gas that may prevent the habitability of inner planets while enabling the habitability of outer ones^{, –}. An atmosphere largely dominated by hydrogen, if cloud-free, should yield prominent spectroscopic signatures in the near-infrared detectable during transits. Observations of the innermost planets have ruled out such signatures. However, the outermost planets are more likely to have sustained such a Neptune-like atmosphere^, . Here, we report observations for the four planets within or near the system’s habitable zone, the circumstellar region where liquid water could exist on a planetary surface …