Erdős's high-girth Steiner system conjecture

Given $n\ge q>r\ge2$, an $(n,q,r)$-Steiner system is a collection of $q$-element subsets of an $n$-element set in which every $r$-element subset lies in exactly one block. A $(j,i)$-configuration is a set of $i$ blocks spanning at most $j$ vertices. Erdős's conjecture. For every integer $g\ge2$, there exists $n_g$ such that for all admissible $n\ge n_g$, there exists an $(n,3,2)$-Steiner system with no $(i+2,i)$-configuration for all $2\le i\le g$. This was fully proved by Kwan, Sah, Sawhney, and Simkin, while earlier work established approximate versions.