Containment probability conjecture for random permutations

Let $k$ be a positive integer. Let $\pi$ be a permutation of length $k$, let $n=1000k^2$, and let $\sigma$ be a uniformly random permutation of length $n$. Containment probability conjecture. The probability that $\sigma$ does not contain $\pi$ satisfies

$$\Pr(\pi\notin\sigma)\leq \exp\bigl(-k^{3/2+o(1)}\bigr).$$

This would imply that a typical permutation of length $1000k^2$ is $k$-universal. The paper gives supporting bounds for very quasirandom and highly structured patterns, while hybrid patterns remain the main obstacle.