Qiao, Park and Koolen's bounded-diameter conjecture for amply regular graphs

Let $G$ be a connected amply regular graph with parameters $(n,d,\alpha,\beta)$, and let $0<\varepsilon<1$ be a real number. Qiao, Park and Koolen's conjecture. There exists a constant $K(\varepsilon)$ such that, if $\beta>\varepsilon d\geq 1$ and $\beta\geq\alpha$, then the diameter of $G$ is at most $K(\varepsilon)$. The conjecture is motivated by the known bound $\operatorname{diam}(G)\leq 7$ when $\beta>d/3\geq 1$ and $\beta\geq\alpha$; whether a bound depending only on $\varepsilon$ holds in the stated generality remains open.