Finiteness conjecture for martingale dimension under sub-Gaussian heat kernel estimates

Let $(X,d,m,\mathcal E,\mathcal F)$ be an MMD space, meaning a metric measure Dirichlet space equipped with the associated diffusion. Suppose it satisfies sub-Gaussian heat kernel estimates and that $m$ is a doubling measure. Martingale-dimension finiteness conjecture. The martingale dimension of the associated diffusion is finite. This asks whether sub-Gaussian heat kernel control and doubling alone guarantee finite martingale dimension; finiteness is unknown in many situations.