  We report a proof that under natural assumptions shock profiles viewed as
heteroclinic travelling wave solutions to a hyperbolically regularized system
of conservation laws are spectrally stable, if the shock amplitude is
sufficiently small. This means that an associated Evans function
$\mathcal{E}:\Lambda\rightarrow\mathbb{C}$ with $\Lambda\subset\mathbb{C}$ an
open superset of the closed right half plane
$\mathbb{H}^+\equiv\{\kappa\in\mathbb{C}:\text{Re}\,\kappa \geq 0\}$, has only
one zero, namely a simple zero at $0$. The result is analogous to the one
obtained in [FS02] and [PZ04] for parabolically regularized systems of
conservation laws, and also distinctly extends findings on hyperbolic
relaxation systems in [PZ04], [MZ09], [Ued09] .
