Drier–Linial conjecture on the Hajós number of random lifts

An $\ell$-lift of $K_n$ is obtained by replacing every vertex of $K_n$ with an independent set of size $\ell$ and every edge with a matching of size $\ell$ between the corresponding fibers. Say that almost all $\ell$-lifts of $K_n$ have a property if all but a vanishing proportion do as $n\to\infty$. The Hajós number of a graph is the largest order of a topological clique occurring as a subgraph. Drier–Linial conjecture. For $\ell\geq\Omega(n)$, almost all $\ell$-lifts of $K_n$ have Hajós number equal to $\Theta(n)$. Drier and Linial established a linear lower bound when $\ell<(1-o(1))n/2$, while the conjecture addresses the larger-lift regime; the supplied text gives no evidence of a resolution.