Bishop comparison conjecture for graphs with nonnegative Bakry–Émery curvature

Let $d_{\max}\in{\mathbb N}$. A graph $G=(V,E)$ has vertex degrees $d_x\leq d_{\max}$ for all $x\in V$ and satisfies $CD(0,\infty)$. Bishop comparison conjecture. There are constants $C_1,C_2>0$, depending only on $d_{\max}$, such that for every $x\in V$ and $r\in{\mathbb N}$,

$$|B_r(x)|\leq C_1(1+r^{C_2}).$$

Thus bounded-degree graphs satisfying $CD(0,\infty)$ should have polynomial volume growth. This is presented as an analogue of Bishop's comparison theorem; the source notes that abelian Cayley graphs provide examples, while the general assertion remains open.