Weak gauge-invariance of dimension two condensate in Yang-Mills theory

We give a formal proof that the space-time average of the vacuum condensate of mass dimension two, i.e., the vacuum expectation value of the squared potential $\mathscr{A}_\mu^2$, is gauge invariant in the weak sense that it is independent of the gauge-fixing condition adopted in quantizing the Yang-Mills theory, at least for the small deformation from the generalized Lorentz and the modified Maximal Abelian gauge. This suggests that the numerical value of the condensate must be the same no matter what gauge-fixing conditions for choosing the representative from the gauge orbit are adopted to measure it. The statement holds even if the Gribov copies exist, by restricting the functional integral inside the Gribov horizon