Steiner triple system hitting-time conjecture

Let $n\equiv 1,3 \pmod 6$, and let $T_1,T_2,\ldots$ be a uniformly random ordering of the triples in $\binom{[n]}{3}$. Steiner triple system hitting-time conjecture. With high probability, the first prefix $T_1,\ldots,T_k$ that covers every $2$-edge at least once contains a Steiner triple system. The conjecture is the corresponding hitting-time version of the proposed sharp threshold and is presented by the source as an open direction.