Curvature and local matchings of conference graphs and extensions

We confirm a conjecture of Bonini et. al. on the precise Lin-Lu-Yau curvature values of conference graphs, i.e., strongly regular graphs with parameters $(4\gamma+1,2\gamma,\gamma-1,\gamma)$, with $\gamma\geq 2$. Our method only depends on the parameter relations and applies to more general classes of amply regular graphs. In particular, we develop a new combinatorial method for showing the existence of local perfect matchings. A key observation is that counting common neighbors leads to useful quadratic polynomials. Our result also leads to an interesting number theoretic consequence on quadratic residues.
Comment: A new sufficient condition (5) where $\alpha\geq \beta$ is added. All comments are welcome!