Logarithmic lower bound conjecture for separating non-contractible cycles

For any map $m$, let $\operatorname{SNC}(m)$ be the size of the shortest separating non-contractible cycle of $m$. Let $q^{(n)}$ be the random uniform high-genus quadrangulation, and let $\theta$ denote the high-genus scaling parameter. Separating-cycle conjecture. There exists a constant $s_\theta$ such that

$$\operatorname{SNC}(q^{(n)})\geq s_\theta\log n$$

whp. This conjecture refines the preceding expectation that the shortest non-contractible cycle is non-separating, while separating non-contractible cycles are substantially larger; proving this logarithmic lower bound remains open.