Logarithmic diameter conjecture for uniform high-genus quadrangulations

Let $q^{(n)}$ be the random uniform quadrangulation in the high-genus regime, with genus parameter determined by $\theta$, and let $\operatorname{diam}(q^{(n)})$ denote its graph diameter. Here, whp means with high probability as $n\to\infty$. Logarithmic diameter conjecture. There exist constants $m_\theta$ and $M_\theta$ such that

$$m_\theta\log(n)\leq \operatorname{diam}(q^{(n)})\leq M_\theta\log(n)$$

whp. The lower bound follows from the paper's planar-neighborhood theorem; the conjecture concerns the matching logarithmic upper bound and is attributed to several people in the community.