  Long-term magma ocean phases on rocky exoplanets orbiting closer to their
star than the runaway greenhouse threshold - the inner edge of the classical
habitable zone - may offer insights into the physical and chemical processes
that distinguish potentially habitable worlds from others. Thermal
stratification of runaway planets is expected to significantly inflate their
atmospheres, potentially providing observational access to the runaway
greenhouse transition in the form of a "habitable zone inner edge
discontinuity" in radius-density space. Here, we use Bioverse, a statistical
framework combining contextual information from the overall planet population
with a survey simulator, to assess the ability of ground- and space-based
telescopes to test this hypothesis.
  We find that the demographic imprint of the runaway greenhouse transition is
likely detectable with high-precision transit photometry for sample sizes
$\gtrsim 100$ planets if at least ~10 % of those orbiting closer than the
habitable zone inner edge harbor runaway climates. Our survey simulations
suggest that in the near future, ESA's PLATO mission will be the most promising
survey to probe the habitable zone inner edge discontinuity. We determine
survey strategies that maximize the diagnostic power of the obtained data and
identify as key mission design drivers: 1. A follow-up campaign of planetary
mass measurements and 2. The fraction of low-mass stars in the target sample.
Observational constraints on the runaway greenhouse transition will provide
crucial insights into the distribution of atmospheric volatiles among rocky
exoplanets, which may help to identify the nearest potentially habitable
worlds.
