Alon's universality conjecture for random permutations

Let $k$ be a positive integer and let $\mathcal{S}_n$ be the set of permutations of $[n]$. A permutation is $k$-universal if it contains every pattern in $\mathcal{S}_k$. For fixed $\varepsilon>0$, consider a uniformly random permutation of length $(1+\varepsilon)k^2/4$. Alon's universality conjecture. With high probability, it is $k$-universal. Here, with high probability means with probability $1-o(1)$ as $k\to\infty$. The conjecture predicts the asymptotically sharp threshold for a random permutation to contain all patterns of length $k$; the paper proves universality for substantially larger lengths but leaves this asymptotic threshold open.