  A non-abelian $SU(2)$ gauge field with a non-minimal Horndeski coupling to
gravity gives rise to a de Sitter solution followed by a graceful exit to a
radiation-dominated epoch. In this Horndeski Yang-Mills (HYM) theory we derive
the second-order action for tensor perturbations on the homogeneous and
isotropic quasi de Sitter background. We find that the presence of the
Horndeski non-minimal coupling to the gauge field inevitably introduces ghost
instabilities in the tensor sector during inflation. Moreover, we also find
Laplacian instabilities for the tensor perturbations deep inside the Hubble
radius during inflation. Thus, we conclude that the HYM theory does not provide
a consistent inflationary framework due to the presence of ghosts and Laplacian
instabilities.
