Elliot–Rödl hypertree embedding conjecture for Steiner triple systems

A hypertree is a connected, simple $3$-uniform hypergraph in which every two vertices are joined by a unique path. A Steiner triple system is a $3$-uniform hypergraph in which every pair of vertices is contained in exactly one edge. For a hypergraph, its order is its number of vertices. Elliot–Rödl's hypertree embedding conjecture. Given $\epsilon>0$, there is $n_0$ such that for any $n\geq n_0$, any hypertree $T$ of order $n$ and any Steiner triple system $S$ of order at least $(1+\varepsilon)n$, $S$ contains $T$ as a subhypergraph. A greedy argument shows that any hypertree with at most $n$ vertices embeds into every Steiner triple system with $2n+1$ vertices; the conjecture asks whether the required order can instead be reduced to any fixed factor greater than $1$ for sufficiently large $n$.