Quadratic convergence conjecture for planar diagonal diffusion matrices with constant trace

Let $\Omega$ be the domain and let $u^{\varepsilon}$ and $u$ denote, respectively, the solution of the rapidly oscillating elliptic equation and its homogenized limit, with data $f,g$ as in the paper. Assume that $A\in C^2(\mathbb{T}^2;\mathcal{S}^2_+)$ has the form

$$A(y)=\operatorname{diag}(a_1(y),a_2(y)),\qquad y\in\mathbb{R}^2,$$

where $a_1,a_2\in C^2(\mathbb{T}^2;(0,1))$ and $a_1+a_2\equiv 1$. Quadratic convergence conjecture. For any choice of $f,g$,

$$\left\|u^{\varepsilon}-u\right\|_{L^{\infty}(\Omega)}=\mathcal{O}(\varepsilon^2)\qquad\text{as }\varepsilon\searrow 0.$$

This conjecture predicts the unexpectedly fast homogenization rate for two-dimensional diagonal diffusion matrices of constant trace; the surrounding discussion notes that the analogous behavior is not observed in dimensions $n\geq 3$.