Discrete–continuum exponent conjecture for separable permutations and cographs

Let $d(1/2)\in[\alpha_*(1/2),\beta^*(1/2)]$ be the exponent from the critical exponent conjecture. Let $\sigma_n$ be a uniform separable permutation of size $n$, and let $G_n$ be a uniform separable cograph of size $n$. Write $\operatorname{LIS}$ for longest increasing subsequence length and $\operatorname{LHS}$ for largest homogeneous-set size.

Discrete–continuum exponent conjecture. With probability tending to $1$ as $n\to\infty$,

$$\operatorname{LIS}(\sigma_n)=n^{d(1/2)+o(1)}\qquad\text{and}\qquad\operatorname{LHS}(G_n)=n^{d(1/2)+o(1)}.$$

The conjecture proposes that the exact polynomial exponent for the Brownian models at parameter $1/2$ also governs the corresponding uniform discrete models. Establishing this transfer is left open in the paper.