  Recently, it has been shown that, in one dimensional fermionic systems, close
to band edges, the zero temperature conductance scales as $1/N^2$, where $N$ is
the system length. This universal subdiffusive scaling of conductance at band
edges has been tied to an exceptional point (EP) of the transfer matrix of the
system that occur at every band edge. Further, in presence of bulk dephasing
probes, this EP has been shown to lead to a counterintuitive superballistic
scaling of conductance, where the conductance increases with $N$ over a finite
but large regime of system lengths. In this work, we explore how these
behaviors are affected by the order of the transfer matrix EP at the band edge.
We consider a one-dimensional fermionic lattice chain with a finite range of
hopping. Depending on the range of hopping and the hopping parameters, this
system can feature band edges which correspond to arbitrarily higher order EPs
of the associated transfer matrix. Using this system we establish in generality
that, in absence of bulk dephasing, surprisingly, the universal $1/N^2$ scaling
of conductance is completely unaffected by the order of the EP. This is despite
the fact that existence of transfer matrix EP is crucial for such behavior. In
presence of bulk dephasing, however, the phase coherence length, the extent of
the superballisitic scaling regime and the exponent of superballistic scaling,
all encode the order of the transfer matrix EP.
