Scaling conjecture for longest subsequences in uniform separable permutations and cographs

Let $calpha(1/2)$ and $X(1/2)$ be the exponent and limiting random variable from the Brownian separable permuton result. Let $coverline{\sigma}_n$ and $coverline{G}_n$ be, respectively, a uniform separable permutation and a uniform separable cograph of size $n$.

Uniform separable scaling conjecture. There is a deterministic constant $c>0$ such that

$$\frac{\operatorname{LIS}(\overline{\sigma}_n)}{n^{\alpha(1/2)}} \xrightarrow[n\rightarrow \infty]{\mathrm{d}} c \cdot X(1/2)$$

and

$$\frac{\operatorname{LIN}(\overline{G}_n)}{n^{\alpha(1/2)}} \xrightarrow[n\rightarrow \infty]{\mathrm{d}} c\cdot X(1/2).$$

The conjecture extends the Brownian separable permuton and cograph results to uniform models; numerical simulations suggest $c\approx 0.901$, and the source notes that this is more precise than an earlier conjecture.