Tightness conjecture for the systole of high-genus quadrangulations

Let $q^{(n)}$ be the random uniform high-genus quadrangulation, and let $\operatorname{syst}(q^{(n)})$ denote the size of its shortest non-contractible cycle. Systole tightness conjecture.

$$\lim_{M\to\infty}\limsup_{n\to\infty}\mathbb{P}\bigl(\operatorname{syst}(q^{(n)})>M\bigr)=0.$$

The paper notes that the systole equals $2$ with positive probability asymptotically and conjectures, in accordance with continuous models, that it remains bounded in probability; the precise limiting behavior is open.