The conjecture on local gradient control for continuously graded composites

Let $S\subset\Omega$ be an open set with closure contained in $\Omega$. Consider continuously graded composites with inclusions having $C^{1,\alpha}$ boundaries, and let $A^{k_j}(x,n_j)$ G-converge to $A^H(x)$. Let $\chi^i_{k_j}(x,n_jx)$ denote the characteristic function of the $i$-th constituent and let $u_j$ and $u^H$ be the corresponding solutions; write ${\mathcal M}^i(\nabla u^H)$ for the associated effective field. Local gradient-control conjecture. One has

$$\limsup_{j\to\infty}\|\chi^i_{k_j}(x,n_jx)\nabla u_j\|_{L^\infty(S)}=\|{\mathcal M}^i(\nabla u^H)\|_{L^\infty(S)}.$$

The conjecture asserts full, rather than merely almost-everywhere outside sets of vanishing measure, control of the local gradients in continuously graded composites with smooth inclusion boundaries. It is motivated by numerical simulations for smooth inclusions and by the homogenization identity, but the supplied text gives no resolution of the conjecture.