The 4/3 diameter conjecture for Morgenstern Ramanujan graphs

Let $q$ be a prime power, let $\mathbb{F}_q$ be the finite field with $q$ elements, and let $g\in\mathbb{F}_q[t]$ satisfy $\gcd(g,t(t-1))=1$. Let $X^{q,g}$ be the associated $q+1$-regular Morgenstern Ramanujan graph. The 4/3 diameter conjecture. If $q$ is odd and $g$ is irreducible, then the diameter of $X^{q,g}$ is bounded from above by

$$\left(\frac{4}{3}+\varepsilon\right)\log_q|X^{q,g}|+O_{\varepsilon}(1).$$

The conjecture is the graph-theoretic problem addressed by the paper and would improve the best currently known general diameter bound for these graphs.