High CO/H2 ratios supports an exocometary origin for a CO-rich debris disk

Over 20 exocometary belts host detectable circumstellar gas, mostly in the form of CO. Two competing theories for its origin have emerged, positing the gas to be primordial or secondary. Primordial gas survives from the belt's parent protoplanetary disk and is therefore H$_2$-rich. Secondary gas is outgassed \textit{in-situ} by exocomets and is relatively H$_2$-poor. Discriminating between these scenarios has not been possible for belts hosting unexpectedly large quantities of CO. We aim to break this gas origin dichotomy \textit{via} direct measurement of H$_2$ column densities in two edge-on CO-rich exocometary belts around $\sim$15 Myr-old A-type stars, constraining the $\frac{\text{CO}}{\text{H}_2}$ ratio and CO gas lifetimes. Observing edge-on belts enables rovibrational absorption spectroscopy against the stellar background. We present near-IR CRIRES+ spectra of HD 110058 and HD 131488 which provide the first direct probe of H$_2$ gas in CO-rich exocometary belts. We target the H$_2$ (v=1-0 S(0)) line at 2223.3 nm and and the $^{12}$CO $v=2\rightarrow0$ rovibrational lines in the range 2333.8-2335.5 nm and derive constraints on column densities along the line-of-sight to the stars. We strongly detect $^{12}$CO but not H$_2$ in the CRIRES+ spectra. This allows us to place $3\sigma$ lower limits on the $\frac{\text{CO}}{\text{H}_2}$ ratios of $> 1.35 \times 10^{-3}$ and $> 3.09 \times 10^{-5}$ for HD 110058 and HD 131488 respectively. These constraints demonstrate that at least for HD 110058, the exocometary gas is compositionally distinct and significantly H$_2$-poor, compared to the $<10^{-4}$ $\frac{\text{CO}}{\text{H}_2}$ ratios typical of protoplanetary disks. We also find H$_2$ alone is unlikely to shield CO over the lifetime of the systems. Overall this suggests that the gas in CO-rich belts is most likely not primordial in origin, supporting the presence of exocometary gas.