  Terrestrial planets in the Habitable Zone of Sun-like stars are priority
targets for detection and observation by the next generation of powerful space
telescopes. Earth's long-term habitability may have been tied to the geological
carbon cycle, a process critically facilitated by plate tectonics. In the
modern Earth, plate motion corresponds to a mantle convection regime called
mobile-lid. The alternate, stagnant-lid regime is found on Mars and Venus,
which may have lacked strong enough weathering feedbacks to sustain surface
liquid water over geological timescales if initially present. Constraining
observational strategies able to infer the most common regime in terrestrial
exoplanets requires quantitative predictions of the atmospheric composition of
planets in either regime. We use endmember models of volcanic outgassing and
crust weathering for the stagnant- and mobile-lid convection regimes, that we
couple to models of atmospheric chemistry and climate, and ocean chemistry to
simulate the atmospheric evolution of these worlds in the Habitable Zone. In
our simulations under the two alternate regimes, we find that the fraction of
planets possessing climates consistent with surface liquid water differ by less
than 10%. Despite this unexpectedly small difference, we predict that a mission
capable of detecting atmospheric CO$_2$ abundance above 0.01 bar in 25
terrestrial exoplanets is extremely likely ($\geq 95$% of samples) to infer the
dominant interior convection regime in that sample with strong evidence (10:1
odds). This offers guidance for the specifications of the Habitable Worlds
Observatory NASA concept mission and other future missions capable of probing
samples of habitable exoplanets.
