Existence of short simple separating cycles in random high-genus triangulations

Fix $\theta\in(0,1/2)$ and a sequence $(g_n)$ such that $g_n/n\rightarrow\theta$. Let $\mathbf{T}_{2n,g_n}$ be a uniform triangulation of genus $g_n$ with $2n$ faces. A simple separating non-contractible cycle is a simple cycle that is separating and non-contractible. The short simple separating cycle conjecture. For every $\theta\in(0,1/2)$, there exists a constant $K$ such that, with high probability, $\mathbf{T}_{2n,g_n}$ contains a simple, non-contractible, separating cycle of length less than $K\log(n)$.

The theorem preceding this conjecture proves only logarithmic upper and lower bounds for the unrestricted separating systole. This conjecture would establish a logarithmically short simple separating cycle in the random model, despite the deterministic existence problem remaining open.