Reduction of structure group of pullbacks of semistable principal bundles

Let C be an irreducible smooth projective curve defined over an algebraically closed field k. Let G be a semisimple linear algebraic group defined over the field k   and P⊂GP⊂G a proper parabolic subgroup. Fix a strictly anti-dominant character χ of P  . Let EGEG be a semistable principal G-bundle over C. If the characteristic of k   is positive, then EGEG is assumed to be strongly semistable. Take any real number ϵ>0ϵ>0. Then there is an irreducible smooth projective curve C˜ defined over k, a nonconstant morphismϕ:C˜→C, and a reduction of structure group EˆP⊂ϕ∗EG of the principal G  -bundle ϕ∗EGϕ∗EG to the subgroup P, such that the following holds:degree(EˆP(χ))degree(ϕ)<ϵ, where EˆP(χ) is the line bundle over C˜ associated to the principal P  -bundle EˆP for the character χ of P.