gas, widening the linewidth [e.g., @Downes1998; @Narayanan2012]. In @French2015, we consider whether a ULIRG-like $\alpha_{\rm CO}$ may be appropriate for post-starburst galaxies. While post-starbursts may be descendants of ULIRGs, we are observing them many dynamical times ($\sim10^6-10^{7.5}$ yr, @Genzel2010) after the starburst phase has ended ($\sim$0.3-1 Gyr ago). We estimate the influence of the stellar potential on increasing the linewidth and lowering $\alpha_{\rm CO}$, and find it is not sufficient to resolve the observed offset between $L^{\prime}($CO$)$ and the low SFRs. Additionally, @Smercina2018 find gas to dust ratios in post-starburst galaxies consistent with nearby galaxies using a Milky-Way like value of $\alpha_{\rm CO}$, indicating significantly lower conversion factors are unlikely. The dense gas conversion factor is similarly uncertain, but to resolve the discrepancies for the post-starburst galaxies in both SFR-$L^{\prime}($CO$)$ and SFR-$L^{\prime}($HCN$)$, some other effect would have to lower $\alpha_{\rm CO}$ without a decrease in $\alpha_{\rm HCN}$.

We test this possibility using [DESPOTIC]{} [@Krumholz2013] to model the change in $\alpha_{\rm CO}$ and $\alpha_{\rm HCN}$ for the typical Milky Way GMC and ULIRG conditions described by @Krumholz2013. ULIRG conditions result in $\alpha_{\rm CO}$ values $\sim5\times$ lower than in Milky Way GMC conditions, as expected. However, $\alpha_{\rm HCN}$ is lowered by the 