The minimum codegree threshold for transversal resolvable Steiner triple systems

Let $n$ be sufficiently large with $n\equiv 3\pmod{6}$, and let $V$ be a common $n$-vertex set. Let $\mathcal{H}=\{H_1,\ldots,H_{n(n-1)/6}\}$ be a collection of $3$-uniform hypergraphs on $V$. A $\mathcal{H}$-transversal resolvable Steiner triple system is a resolvable Steiner triple system whose triples can be assigned injectively to the hypergraphs so that each assigned triple belongs to its assigned hypergraph. The transversal resolvable Steiner triple system conjecture. There are constants $n_0$ and $C$ such that, whenever $n\ge n_0$ and every $H_i$ has minimum codegree at least $(3/4)n+C$, the collection $\mathcal{H}$ contains a $\mathcal{H}$-transversal resolvable Steiner triple system on $V$. The existence of resolvable systems explains the congruence condition $n\equiv3\pmod6$; the minimum-codegree assertion remains open.