  The recent LHC discovery of a Higgs-like resonance at 126 GeV suggests that
the minimal supersymmetric standard model must be modified in order to preserve
naturalness. A simple extension is to include a singlet superfield and consider
the scale-invariant NMSSM, whose renormalizable superpotential contains no
dimensionful parameters. This extension not only solves the \mu-problem, but
can easily accommodate a 126 GeV Higgs. We study the naturalness of the
scale-invariant NMSSM taking into account the recent constraints from LHC
searches, flavor physics and electroweak precision tests. We show that
TeV-scale stop masses are still allowed in much of the parameter space with 5%
tuning for a low messenger scale of 20 TeV, split families (with
third-generation sleptons decoupled) and Higgs-singlet coupling \lambda of
order one. For larger values of the Higgs-singlet coupling, which can relieve
the tuning in the Higgs VEV, an additional tuning in the Higgs mass limits
increasing the (lightest) stop mass beyond 1.2 TeV, the gluino mass above 3
TeV, and electroweak charginos and neutralinos beyond 400 GeV for a combined
tuning better than 5%. This implies that the natural region of parameter space
for the scale-invariant NMSSM will be fully explored at the 14 TeV LHC.
