Polynomial growth and diffusive random walk for non-negatively curved graphs

Let $G=(V,E)$ be a graph, let $dgeq 1$, and suppose that $G$ has non-negative Ollivier--Ricci curvature and degrees bounded by $d$. Write $B(x,r)$ for the ball of radius $r$ centered at $x$, and let $X_n$ be the random walk started at $x$, with graph distance denoted by $d(X_0,X_n)$. Polynomial-growth and diffusive-walk conjecture. For each $d\geq 1$ there exists a constant $C_d$ such that

$$\#B(x,r)\leq r^{C_d} \qquad \text{and} \qquad \mathbf{E}_x\!\left[d(X_0,X_n)^2\right]\leq C_d n$$

for every $n,r\geq 1$ and $x\in V$. This would strengthen the paper's subexponential-growth and near-diffusive estimates by giving polynomial growth of uniformly bounded dimension and a genuinely diffusive displacement bound, uniformly over the root vertex; the conjecture is presented as open.