Optimal stochastic homogenization rate under the CFS condition

Let $\mathbb{P}$ satisfy the condition $\mathsf{CFS}(\beta,\Psi)$, where $\beta\in(\frac12,1)$, and let ${\mathbf{b}}_r(x)$ and $\bar{\mathbf{a}}$ denote the corresponding rescaled coefficient field and homogenized coefficient. For $r\geq2$, write $\mathcal{O}_{\Psi}^c$ for the stochastic integrability notation used in the paper. Conjecture on the optimal rate. There exist constants $C(\mathrm{data})<\infty$ and $c(d,\beta)\in(0,1)$ such that

$${\mathbf{b}}_r(x)-\bar{\mathbf{a}}=\mathcal{O}_{\Psi}^c\bigl(Cr^{-\frac d2(1-\beta)}\bigr).$$

This extends the preceding proposition, which establishes the same rate under the stronger lower bound $\beta\geq\frac1d(d+1-\sqrt{d+1})$. The conjecture concerns optimal quantitative stochastic homogenization estimates for the full range $\beta\in(\frac12,1)$; the precise stochastic integrability and rate in this range remain open.