  The ultracool dwarf star TRAPPIST-1 hosts seven Earth-size transiting
planets, some of which could harbour liquid water on their surfaces. UV
observations are essential to measure their high-energy irradiation, and to
search for photodissociated water escaping from their putative atmospheres. Our
new observations of TRAPPIST-1 Ly-$\alpha$ line during the transit of
TRAPPIST-1c show an evolution of the star emission over three months,
preventing us from assessing the presence of an extended hydrogen exosphere.
Based on the current knowledge of the stellar irradiation, we investigated the
likely history of water loss in the system. Planets b to d might still be in a
runaway phase, and planets within the orbit of TRAPPIST-1g could have lost more
than 20 Earth oceans after 8 Gyr of hydrodynamic escape. However, TRAPPIST-1e
to h might have lost less than 3 Earth oceans if hydrodynamic escape stopped
once they entered the habitable zone. We caution that these estimates remain
limited by the large uncertainty on the planet masses. They likely represent
upper limits on the actual water loss because our assumptions maximize the
XUV-driven escape, while photodissociation in the upper atmospheres should be
the limiting process. Late-stage outgassing could also have contributed
significant amounts of water for the outer, more massive planets after they
entered the habitable zone. While our results suggest that the outer planets
are the best candidates to search for water with the JWST, they also highlight
the need for theoretical studies and complementary observations in all
wavelength domains to determine the nature of the TRAPPIST-1 planets, and their
potential habitability.
