Nine-vertex 2-core conjecture for Steiner triple systems

A Steiner triple system is a 3-uniform hypergraph in which every pair of vertices lies in exactly one edge. A 2-core is a hypergraph with minimum degree $2$.

Nine-vertex 2-core conjecture. Every sufficiently large Steiner triple system contains a 2-core on nine vertices.

The conjecture is motivated by observations that all Steiner triple systems of size $21$ with a non-trivial automorphism group contain a $3\times 3$ grid, as well as by the Brown–Erdős–Sós conjecture and known results that every Steiner triple system contains a core on at most ten vertices. Its status is open.