$H_0$ tension or $M$ overestimation?

There is a strong discrepancy between the value of the Hubble parameter $H_0^P$ obtained from large scale observations such as the Planck mission, and the small scale value $H_0^R$, obtained from low redshift supernovae (SNe). The value of the absolute magnitude $M^{Hom}$ used as prior in analyzing observational data is obtained from low-redshift SNe, assuming a homogeneous Universe, but the distance of the anchors used to calibrate the SNe to obtain M would be affected by a local inhomogeneity, making it inconsistent to test the Copernican principle using $M^{Hom}$, since M estimation itself is affected by local inhomogeneities. We perform an analysis of the luminosity distance of low redshift SNe, using different values of M, $\{M^P,M^R\}$, corresponding to different values of $H_0$, $\{H_0^P,H_0^R\}$, obtained from the model independent consistency relation between $H_0$ and M which can be derived from the definition of the distance modulus. We find that the value of M can strongly affect the evidence of a local inhomogeneity. We analyze data from the Pantheon catalog, finding no significant statistical evidence of a local inhomogeneity using the parameters $\{M^R,H_0^R\}$, confirming previous studies, while with $\{M^P,H_0^P\}$ we find evidence of a small local void, which causes an overestimation of $M^R$ with respect to $M^P$. An inhomogeneous model with the parameters $\{M^P,H_0^P\}$ fits the data better than a homogeneous model with $\{M^R,H_0^R\}$, resolving the apparent $H_0$ tension. Using $\{M^P,H_0^P\}$, we obtain evidence of a local inhomogeneity with a density contrast $-0.140 \pm 0.042 $, extending up to a redshift of $z_v =0.056 \pm 0.0002$, in good agreement with recent results of galaxy catalogs analysis (Wong et al. in The local hole: a galaxy under-density covering 90 mpc, 2021).