Non-expansion conjecture for bounded-degree graphs with nonnegative Bakry–Émery curvature

Let $d\in{\mathbb N}$. Non-expansion conjecture. No infinite family of finite, increasing, $d$-regular graphs satisfying $CD(0,\infty)$ can be a family of expander graphs. This is stated as a direct consequence of the preceding polynomial volume-growth conjecture, since increasing regular graphs with uniformly bounded polynomial volume growth should have Cheeger constants tending to zero.