The Bakry–Émery curvature Bonnet–Myers inequality conjecture for regular graphs

Let $G=(V,E)$ be a finite connected $D$-regular graph with diameter $L$. The normalized Bakry–Émery curvature at infinity at a vertex $x$ is denoted by ${\mathcal K}^{\rm n}_{G,x}(\infty)$.

Bakry–Émery curvature Bonnet–Myers conjecture. One has

$$\inf_{x \in V} {\mathcal K}^{\rm n}_{G,x}(\infty) \le \frac{1}{D} + \frac{1}{L}.$$

The conjecture would extend the sharp bound known for Bonnet–Myers sharp graphs to all finite connected regular graphs. The source notes that no known $D$-regular graph of diameter $L$ violates this inequality; its status is otherwise unresolved.