  We search for asymptotic safety in a Yukawa system with a chiral
$U(N_L)_L\otimes U(1)_R$ symmetry, serving as a toy model for the
standard-model Higgs sector. Using the functional RG as a nonperturbative tool,
the leading-order derivative expansion exhibits admissible non-Ga\ssian
fixed-points for $1 \leq N_L \leq 57$ which arise from a conformal threshold
behavior induced by self-balanced boson-fermion fluctuations. If present in the
full theory, the fixed-point would solve the triviality problem. Moreover, as
one fixed point has only one relevant direction even with a reduced hierarchy
problem, the Higgs mass as well as the top mass are a prediction of the theory
in terms of the Higgs vacuum expectation value. In our toy model, the fixed
point is destabilized at higher order due to massless Goldstone and fermion
fluctuations, which are particular to our model and have no analogue in the
standard model.
