Weak Hessian for solutions to certain quasilinear elliptic PDEs

Question

In Chapter 4 of their famous treatise Linear and quasilinear elliptic equations, Ladyzhenskaya and Uraltseva deal with equations of the form $$\sum_{i=1}^n\frac{d}{dx_i}a_i(x,u,\nabla u)+a(x,u,\nabla u)=0,$$ where $\sum_i a_i(x,u,p)p_i$ grows like $|p|^m$ for some $1 < m < \infty$ (plus other technical assumptions). For the sake of the discussion, take the PDE $$\operatorname{div}((1+|\nabla u|^2)^{m/2-1}\nabla u)=0$$ on the unit ball $B^n$, with $1 < m < 2$. A weak solution $u\in W^{1,m}$ is locally bounded (and Hölder). To get an $L^\infty$ bound on $\nabla u$, the authors use a technique similar to Moser iteration, assuming however that $u\in W^{2,1}_{loc}$ and $$\int(1+|\nabla u|)^{m-2}|\nabla^2 u|^2<\infty\qquad( * )$$ (locally). Later, in Section 4.5, in order to show that this assumption always holds they use the finite difference method. Letting $\tau_h u(x):=u(x+he_k)$ and $u_{(k)}:=\frac{\tau_h u-u}{h}$, for a given $k=1,\dots,n$, they get $$\begin{align}&\int(1+|\nabla u|+|\nabla\tau_h u|)^{m-2}\sum_i|u_{(k)x_i}|^2 \\ &+\int(1+|\nabla u|+|\nabla\tau_h u|)^{m}|u_{(k)}|^2\zeta^2 \\ &\le C\int(1+|\nabla u|+|\nabla\tau_h u|)^{m-2}|u_{(k)}|^2|\nabla\zeta|^2\end{align}$$ for any cut-off function $\zeta\in C^\infty_c(B^n)$. This is (5.10) in the book. Morally, passing to the limit $h\to 0$ this should correspond to $(*)$. While this is enough for $m\ge 2$, since the right-hand side is uniformly bounded as $h\to 0$, I cannot see how one concludes for $1 < m < 2$. Does anybody know how to deal with this issue?