  Recent studies have provided evidence for the existence of new asymptotically
free trajectories in non-Abelian particle models without asymptotic symmetry in
the high-energy limit. We extend these results to a general ${\rm SU}(N_{\rm
L})\times {\rm SU}(N_{\rm c})$ Higgs-Yukawa model that includes the non-Abelian
sector of the standard model, finding further confirmation for such scenarios
for a wide class of regularizations that account for threshold behavior
persisting to highest energies. We construct these asymptotically free
trajectories within conventional $\overline{\text{MS}}$ schemes and systematic
weak coupling expansions. The existence of these solutions is argued to be a
scheme-independent phenomenon, as demonstrated for mass-dependent schemes based
on general momentum-space infrared regularizations. A change of scheme induces
a map of the theory's coupling space onto itself, which in the present case
also translates into a reparametrization of the space of asymptotically free
solutions.
