  The Lema\^{\i}tre -- Tolman model with $\Lambda = 0$ and constant bang time
that imitates the luminosity distance -- redshift relation of the $\Lambda$CDM
model using the energy function $E$ alone contains shell crossings. In this
paper, the location in spacetime and the consequences of existence of the
shell-crossing set (SCS) are investigated. The SCS would come into view of the
central observer only at $t \approx 1064 T$ to the future from now, where $T$
is the present age of the Universe, but would not leave any recognizable trace
in her observations. Light rays emitted near to the SCS are blueshifted at the
initial points, but the blueshift is finite, and is overcompensated by
later-induced redshifts if the observer is sufficiently far. The local
blueshifts cause that $z$ along a light ray is not a monotonic function of the
comoving radial coordinate $r$. As a consequence, the angular diameter distance
$D_A$ and the luminosity distance $D_L$ from the central observer fail to be
functions of $z$; the relations $D_A(z)$ and $D_L(z)$ are multiple-valued in a
vicinity of the SCS. The following quantities are calculated and displayed: (1)
The distribution of mass density on a few characteristic hypersurfaces of
constant time; some of them intersect the SCS. (2) The distribution of density
along the past light cone of the present central observer. (3) A few light
cones intersecting the SCS at characteristic instants. (4) The redshift
profiles along several light cones. (5) The extremum-redshift hypersurface. (6)
The $D_A(z)$ and $D_L(z)$ relations. (7) The last scattering time and its
comparison with the $\Lambda$CDM last scattering epoch.
