  We assess the question whether the SLED (Supersymmetric Large Extra
Dimensions) model admits phenomenologically viable solutions with 4D maximal
symmetry. We take into account a finite brane width and a scale invariance (SI)
breaking dilaton-brane coupling, both of which should be included in a
realistic setup. Provided that the microscopic size of the brane is not tuned
much smaller than the fundamental bulk Planck length, we find that either the
4D curvature or the size of the extra dimensions is unacceptably large. Since
this result is independent of the dilaton-brane couplings, it provides the
biggest challenge to the SLED program.
  In addition, to clarify its potential with respect to the cosmological
constant problem, we infer the amount of tuning on model parameters required to
obtain a sufficiently small 4D curvature. A first answer was recently given in
[arXiv:1508.01124], showing that 4D flat solutions are only ensured in the SI
case by imposing a tuning relation, even if a brane-localized flux is included.
In this companion paper, we find that the tuning can in fact be avoided for
certain SI breaking brane-dilaton couplings, but only at the price of worsening
the phenomenological problem.
  Our results are obtained by solving the full coupled Einstein-dilaton system
in a completely consistent way. The brane width is implemented using a
well-known ring regularization. In passing, we note that for the couplings
considered here the results of [arXiv:1508.01124] (which only treated
infinitely thin branes) are all consistently recovered in the thin brane limit,
and how this can be reconciled with the concerns about their correctness,
recently brought up in [arXiv:1509.04201].
