  In this paper, we present a partial result on the global well-posedness of
the Cauchy problem for the Einstein-Yang-Mills system in the constant mean
extrinsic curvature spatial harmonic and generalized Coulomb gauges as
introduced in [Mondal, arXiv:2112.14273]. We give a small-data global existence
theorem for a family of $n+1$ dimensional spacetimes with $n\geq4$, utilizing
energy arguments presented in [Andersson and Moncrief, arXiv:0908.0784]. We
observe that these energy arguments will fail for $n=3$ due to the conformal
invariance of the $3+1$ Yang-Mills equations and present a gauge-covaraiant
formulation of the Einstein-Yang-Mills system in $3+1$ dimensions to show that
an energy argument cannot be used to prove the global well-posedness result,
regardless of the choice of gauge.
