A cosmological underdensity does not solve the Hubble tension

A potential solution to the Hubble tension is hypothesis that Milky Way located near center of a matter underdensity. We model this scenario through Lema\^itre-Tolman-Bondi formalism with inclusion cosmological constant ($\Lambda$LTB) and consider generalized Gaussian parametrization for density profile. constrain underdensity background cosmology combination data sets: Pantheon Sample type Ia supernovae (both full catalogue redshift-binned version it), collection baryon acoustic oscillations points distance priors extracted from latest Planck release. The analysis binned suggests preference $-13 \%$ drop size approximately 300 Mpc, interestingly matching prediction so-called KBC void already identified on basis independent analyses using galaxy distributions. constraints obtained are instead compatible homogeneous we interpret radically different result as cautionary tale about bias introduced by employing supernova set. quantify level improvement analyzing B-band absolute magnitude supernovae, which provides calibration local measurements $H_0$. Since no significant difference observed respect an analogous fit performed standard $\Lambda$CDM cosmology, conclude presence does not resolve significantly degrade current