Quantitative martingale-dimension conjecture under sub-Gaussian heat kernel estimates

Let $(X,d,m,\mathcal E,\mathcal F)$ be an MMD space. Suppose it satisfies sub-Gaussian heat kernel estimates with volume growth exponent $\alpha$ and walk dimension $\beta$, and let $d_m$ denote the martingale dimension. Quantitative martingale-dimension conjecture. One has

$$d_m \leq \frac{2\alpha}{\beta}.$$

This is presented as a quantitative version of the finiteness conjecture, supported by the paper's main result and a bound of Hino. The conjecture remains open in the stated generality.