  Efforts to spectrally characterize the atmospheric compositions of temperate
terrestrial exoplanets orbiting M-dwarf stars with the James Webb Space
Telescope (JWST) are now underway. Key molecular targets of such searches
include O$_2$ and CO, which are potential indicators of life. Recently, it was
proposed that CO$_2$ photolysis generates abundant ($\gtrsim0.1$ bar) abiotic
O$_2$ and CO in the atmospheres of habitable M-dwarf planets with CO$_2$-rich
atmospheres, constituting a strong false positive for O$_2$ as a biosignature
and further complicating efforts to use CO as a diagnostic of surface biology.
Importantly, this implied that TRAPPIST-1e and TRAPPIST-1f, now under
observation with JWST, would abiotically accumulate abundant O$_2$ and CO, if
habitable. Here, we use a multi-model approach to re-examine photochemical
O$_2$ and CO accumulation on planets orbiting M-dwarf stars. We show that
photochemical O$_2$ remains a trace gas on habitable CO$_2$-rich M-dwarf
planets, with earlier predictions of abundant O$_2$ and CO due to an
atmospheric model top that was too low to accurately resolve the unusually-high
CO$_2$ photolysis peak on such worlds. Our work strengthens the case for O$_2$
as a biosignature gas, and affirms the importance of CO as a diagnostic of
photochemical O$_2$ production. However, observationally relevant false
positive potential remains, especially for O$_2$'s photochemical product O$_3$,
and further work is required to confidently understand O$_2$ and O$_3$ as
biosignature gases on M-dwarf planets.
