  In this paper we formulate and prove a gauge equivalence for unitary
connections and skew-Hermitian Higgs fields of suitable regularity that are
mapped to the same function under the non-abelian X-ray transform on
nontrapping asymptotically hyperbolic spaces with negative curvature and no
nontrivial twisted conformal Killing tensor fields with certain regularity. If
one furthermore fixes such a connection with zero curvature, a corollary
provides an injectivity result for the non-abelian X-ray transform over
skew-Hermitian Higgs fields.
