  Biosignature detection in the atmospheres of Earth-like exoplanets is one of
the most significant and ambitious goals for astronomy, astrobiology, and
humanity. Molecular oxygen is among the strongest indicators of life on Earth,
but it will be extremely difficult to detect via transmission spectroscopy. We
used the Bioverse statistical framework to assess the ability to probe
Earth-like O$_{\mathrm{2}}$ levels on hypothetical nearby habitable zone
exoplanets (EECs) using direct imaging and high-resolution spectroscopy on the
Giant Magellan Telescope (GMT) and the Extremely Large Telescope (ELT). We
found that O$_{\mathrm{2}}$ could be probed on up to $\sim$5 and $\sim$15 EECs
orbiting bright M dwarfs within 20 pc in a 10-year survey on the GMT and ELT,
respectively. Earth-like O$_{\mathrm{2}}$ levels could be probed on four known
super-Earth candidates, including Proxima Centauri b, within about one week on
the ELT and a few months on the GMT. We also assessed the ability of the ELT to
test the habitable zone oxygen hypothesis $\unicode{x2013}$ that habitable zone
Earth-sized planets are more likely to have O$_{\mathrm{2}}$ $\unicode{x2013}$
within a 10-year survey using Bioverse. Testing this hypothesis requires either
$\sim$1/2 of the EECs to have O$_{\mathrm{2}}$ or $\sim$1/3 if $\eta_{\oplus}$
is large. A northern hemisphere large-aperture telescope, such as the Thirty
Meter Telescope (TMT), would expand the target star pool by about 25%, reduce
the time to probe biosignatures on individual targets, and provide an
additional independent check on potential biosignature detections.
