Partial homogenization conjecture for adaptive weight systems

Consider the SDE system with coefficients satisfying Assumption~ and $\epsilon=1$. For $k\in\mathbb{N}$, let deterministic initial data $(X^{(k)}_i(0))_{i\in[N_k]}$ and $(w^{(k)}_{i,j}(0))_{i,j\in[N_k]}$ satisfy Assumption~, and suppose

$$\lim_{k\to\infty}\sup_{k_1,k_2\geq k}\gamma_{W_1,\square}\bigg(\Big((X^{(k_1)}_i(0))_{i\in[N_{k_1}]},(w^{(k_1)}_{i,j}(0))_{i,j\in[N_{k_1}]},[N_{k_1}]\Big),\Big((X^{(k_2)}_i(0))_{i\in[N_{k_2}]},(w^{(k_2)}_{i,j}(0))_{i,j\in[N_{k_2}]},[N_{k_2}]\Big)\bigg)=0.$$

Partial homogenization conjecture. If the corresponding solutions of are denoted by $(\bm{X}^{(k)}_i(t))_{i\in[N_k]}$ and $(\bm{w}^{(k)}_{i,j}(t))_{i,j\in[N_k]}$, then

$$\lim_{t\to\infty}\limsup_{k\to\infty}\sup_{k_1,k_2\geq k}\mathbb{E}\bigg[\delta_{W_1,\square}\bigg(\Big((\bm{X}^{(k_1)}_i(t))_{i\in[N_{k_1}]},(\bm{w}^{(k_1)}_{i,j}(t))_{i,j\in[N_{k_1}]},[N_{k_1}]\Big),\Big((\bm{X}^{(k_2)}_i(t))_{i\in[N_{k_2}]},(\bm{w}^{(k_2)}_{i,j}(t))_{i,j\in[N_{k_2}]},[N_{k_2}]\Big)\bigg)\bigg]=0.$$

The claim predicts that initial data asymptotically indistinguishable in the weaker metric become indistinguishable in the stronger metric at long times, expressing partial homogenization of the weight structures.