  In this article we study the Conrady-Hnybida extension of the Lorentzian
Engle-Pereira-Rovelli-Livine spin-foam model, which admits time-like cells
rather than just space-like ones. Our focus is on the asymptotic analysis of
the model's vertex amplitude.
  We propose a new parametrization for states associated to time-like 3-cells,
from which we derive a closed-form expression for their amplitudes. This allows
us to revisit the conditions under which critical points of the amplitudes
occur, and we find Regge-like geometrical critical points in agreement with the
literature. However, we find also evidence for non-geometrical points which are
not dynamically suppressed without further assumptions; the model then does not
strictly asymptote to the Regge action, contrary to what one would expect. We
moreover prove Minkowski and rigidity theorems for Minkowskian polyhedra,
extending the asymptotic analysis to non-simplicial spin-foams.
