Erdős's conjecture on sparse Steiner triple systems

For an integer $e\ge 2$, an $(n,3,2)$-design is an $e$-sparse Steiner triple system if it is simultaneously $\mathscr{G}_3(i+2,i)$-free for every $2\le i\le e$. Erdős's conjecture. For every fixed integer $e\ge 2$, there exists $n_0=n_0(e)$ such that an $n$-vertex $e$-sparse Steiner triple system exists for every $n\ge n_0$ satisfying

$$n\equiv 1,3\pmod 6.$$

The paper reports substantial recent progress, including constructions of $e$-sparse packings, but the stated Steiner-system existence assertion is not presented as fully resolved.