Fixed-pattern containment conjecture at the universality threshold

Let $\mathcal{S}_k$ denote the permutations of $[k]$. Fix $\varepsilon>0$ and set $n=(1+\varepsilon)k^2/4$. For $\pi\in\mathcal{S}_k$, let $\sigma$ be a uniformly random permutation of length $n$. Fixed-pattern containment conjecture. With high probability, $\sigma$ contains $\pi$. The statement is a weakening of Alon's universality conjecture because it fixes one pattern rather than requiring simultaneous containment of every pattern in $\mathcal{S}_k$. The source explicitly describes it as an apparently open problem.