Extremal-size conjecture for regular graphs with positive Lin–Lu–Yau curvature

Let $\mathcal{M}_{d}$ denote the maximum number of vertices of a connected $d$-regular graph with positive Lin–Lu–Yau curvature. For even degree, write $d\in2\mathbb{N}$. Extremal-size conjecture. For every $d\in2\mathbb{N}$,

$$\mathcal{M}_{d}=\sqrt{5}^{d}.$$

The Cartesian power $(C_{5})^{\square n}$ supplies the lower bound $\mathcal{M}_{d}\geq\sqrt{5}^{d}$ for even $d$ and has constant minimal positive Lin–Lu–Yau curvature. Determining the exact values of $\mathcal{M}_{d}$ for $d\geq4$ remains open.