Hutchcroft–Münch conjecture on polynomial growth and diffusive displacement

Let $G=(V,E)$ be a graph with non-negative Ollivier–Ricci curvature and degrees bounded by $d<\infty$. Write $B(x,r)$ for the ball of radius $r$ centred at $x$, let $\operatorname{Vol}(B(x,r))$ denote its volume, and let $(X_n)$ be random walk started at $x$. Hutchcroft–Münch conjecture. There exists a constant $C_d$ such that

$$\operatorname{Vol}(B(x,r)) \leq r^{C_d} \qquad\text{and}\qquad \mathbb{E}_x\left[d(x,X_n)^2\right] \leq C_d n$$

for every $n,r\geq 1$ and $x\in V$. This conjecture asks for uniform polynomial volume growth and diffusive random-walk displacement under bounded degree and non-negative Ollivier–Ricci curvature, strengthening the known averaged subexponential growth and near-diffusive displacement estimates. The uniform, rootwise bounds remain open.