Elucidating cosmological model dependence with $H_0$

We observe that the errors on the Hubble constant $H_0$, a universal parameter in any FLRW cosmology, can be larger in specific cosmological models than Gaussian Processes (GP) data reconstruction. We comment on the prior mean function and trace the smaller GP errors to stronger correlations, which we show precludes all well studied dynamical dark energy models. Our analysis suggests that "cosmological model independence", especially in the statement of Hubble tension, has become a misnomer.