Large expander conjecture for high-genus triangulations

Let $g_n$ satisfy

$$\frac{g_n}{n}\to\theta\in(0,1/2),$$

and let $\mathbf{T_{n,g_n}}$ be a uniform triangulation of genus $g_n$ with $3n$ edges. An induced subgraph is a $\kappa$-expander if every set of vertices has at least $\kappa$ times its size in edges leaving it. Large expander conjecture. For every $\varepsilon>0$, there exists a $\kappa>0$ depending only on $\varepsilon$ and $\theta$ such that, with high probability, $\mathbf{T_{n,g_n}}$ contains an induced subgraph $G_n$ with at least $(1-\varepsilon)3n$ edges that is a $\kappa$-expander. This conjecture extends the paper's theorem for uniform unicellular maps to uniform triangulations, and concerns the global geometric structure of high-genus maps. It is presented as an ambitious open problem.