Realizability conjecture for volume growth, escape time and martingale dimensions

Let $b1 b2$ and $d_m$ satisfy

$$b1 \in [1,\infty),\qquad b2 \in [2,\infty),\qquad d_m \in \mathbb{N},$$

and

$$2 \leq b2 \leq b1+1,\qquad 1 \leq d_m \leq \frac{2b1}{b2}.$$

Here $b1$ is the volume growth exponent, $b2$ is the escape time exponent, and $d_m$ is the martingale dimension, also called the martingale index. Realizability conjecture. For every such $b1$, $b2$, and $d_m$, there exists a symmetric diffusion process on a metric measure space satisfying a full sub-Gaussian heat kernel estimate with volume growth exponent $b1$, escape time exponent $b2$, and martingale dimension (or index) $d_m$. The conjecture concerns which triples of exponents and dimensions can occur; the supplied text gives no resolution, so its status is open.