Hausdorff-dimension bound for martingale dimension under sub-Gaussian estimates

Let $(X,d,m,\mathcal E,\mathcal F)$ be an MMD space, and let $d_m$ denote the martingale dimension of its associated diffusion. Suppose that $(X,d,m,\mathcal E,\mathcal F)$ satisfies sub-Gaussian heat kernel estimates. Hausdorff-dimension bound conjecture. Then

$$d_m \leq \dim_H(X,d).$$

This is stated as a weaker version of the quantitative martingale-dimension conjecture, using that the walk dimension is at least two. It remains open in the generality stated.