Pro-l fundamental groups of generically ordinary semi-stable fibrations with low slope

Let k be an algebraically closed field of characteristic p > 0 and l a prime that is distinct from p. Let f : S \rightarrow C be a generically ordinary, semi-stable fibration of a projective smooth surface S to a projective smooth curve C over k. Let F be a general fibre of f, which is a smooth curve of genus g \geq 2. We assume that f is generically strongly l-ordinary, by which we mean that every cyclic etale covering of degree l of the generic fibre of f is ordinary. Suppose that f is not locally trivial and is relatively minimal. Then deg f*\omegaS/C > 0, where \omegaS/C is the sheaf associated to the relative canonical divisor KS/C = KS − f*KC. Hence the slope of f,\lambda( f ) = K2 S/C/deg f*\omegaS/C is well-defined. Consider the push-out square \pi1(F) \rightarrow \pi1(S) \rightarrow \Pi(C) \rightarrow 1 \downarrow \Pi where \pi1 is the algebraic fundamental group and \pil1 is the pro-l fundamental group. When f is non-hyperelliptic and \lambda(f) < 4, we show that the morphism \pil1(F)\rightarrow /alpha\Pi is trivial