  We investigate Lema\^itre-Tolman-Bondi (LTB) models, whose early time
evolution and bang time are homogeneous and the distance - redshift relation
and local Hubble parameter are inherited from the $\Lambda$CDM model. We show
that the obtained LTB models and the $\Lambda$CDM model predict different
relative local expansion rates and that the Hubble functions of the models
diverge increasingly with redshift. The LTB models show tension between low
redshift baryon acoustic oscillation and supernova observations and including
Lyman-$\alpha$ forest or cosmic microwave background observations only
accentuates the better fit of the $\Lambda$CDM model compared to the LTB model.
The result indicates that additional degrees of freedom are needed to explain
the observations, for example by renouncing spherical symmetry, homogeneous
bang time, negligible effects of pressure, or the early time homogeneity
assumption.
