Representations of surface groups with universally finite mapping class group orbit

Let Sigma(g,n) be the orientable genus g surface with n punctures, where 2 - 2g - n < 0. Let rho : pi(1) (Sigma(g,n)) -> GL(m) (C) be a representation. Suppose that for each finite covering map f : Sigma(g'),(n') -> Sigma(g,n), the orbit of (the isomorphism class of) f* (rho) under the mapping class group MCG(Sigma(g',n')) of Sigma(g',n') is finite. Then we show that rho has finite image. The result is motivated by the Grothendieck-Katz p-curvature conjecture, and gives a reformulation of the p-curvature conjecture in terms of isomonodromy.