  We investigate Planck scale boundary conditions on Higgs quartic interactions
and their $\beta$-functions in the SM augmented by an additional complex
scalar. We use renormalisation group running at two-loops, and include both
theoretical and experimental constraints. We find that the boundary condition
$\lambda=\beta_\lambda=0$ at the Planck scale is compatible with the current
Higgs and top mass measurements, but requires additional scalars lighter than
about $600\,$GeV.
