Asymptotic conjecture for size-enumerated self-dual interval orders

Let $r_n$ be the number of self-dual interval orders of size $n$, and let $q_n$ be the number of primitive self-dual interval orders of size $n$. Size asymptotic conjecture.

$$r_n=\left(\gamma+O(n^{-1/2})\right)\sqrt{n}\left(\frac{\delta n}{e}\right)^{n/2}2^{\sqrt{\delta n}},$$

with $\gamma\approx 1.361951039\dots$ and $\delta=\frac{6}{\pi^2}$, and

$$\lim_{n\to\infty}\frac{q_n}{r_n}=\frac{1}{2}e^{-\pi^2/12}.$$

These are numerical conjectures for the coefficient asymptotics of self-dual interval orders counted by size. The source identifies analogous asymptotic enumeration as its main general open problem and does not give a resolution.