The hitting-time conjecture for Steiner systems in random hypergraph processes

Let $K_k^n$ be the complete $k$-uniform hypergraph on $n$ vertices, and expose its edges one by one in a uniformly random order to obtain the random hypergraph process. A facet is a $(k-1)$-subset of vertices, and a facet is covered when it is contained in at least one exposed edge. Steiner-system hitting-time conjecture. As $n\to\infty$ with $k$ fixed, asymptotically almost surely a Steiner system appears at the very moment that all facets are covered. This would identify the sole obstruction to a Steiner system in the process with the presence of uncovered facets, extending the corresponding graph-process phenomenon; the analogous assertion for perfect matchings in hypergraphs is noted as open.