  The standard model of modern cosmology might be cracked by the recent
persistent hot debate on the Hubble-constant ($H_0$) tension, which manifests
itself as the sound-horizon ($r_s$) tension or absolute-magnitude ($M_B$)
tension if deeming the origin of the Hubble tension from modifying the early or
late Universe, respectively. In this Letter, we achieve a fully
model-independent constraint (fitting a model-independent global
parameterization to a model-independent inverse distant ladder with a
model-independent high-redshift calibration) on late-time models with strong
evidence against homogeneous new physics over the $\Lambda$-cold-dark
($\Lambda$CDM) model. Further using this model-independent constraint to
calibrate sufficiently local supernovae with corresponding late-time models
extrapolated below the homogeneity scale, we find surprisingly that, although
both $H_0$ tension and $M_B$ tension are absent in our local Universe, a
combination of $H_0$ and $M_B$ as the intercept $a_B$ of the magnitude-redshift
relation exhibits $3\sim 7\sigma$ tension even for the $\Lambda$CDM model. This
$a_B$ tension seems to call for local-scale inhomogeneous new physics disguised
as local observational systematics.
