Bollobás–Erdős-type conjecture for the high-girth triple-process

Let $\ell \ge 4$. Let $m$ be the total number of steps in the high-girth triple-process that produces a partial Steiner system with girth greater than $\ell$. High-girth process conjecture. With probability $1-o(1)$,

$$\binom{n}{2}-3m=n^{3/2+o(1)}.$$

This predicts that the high-girth process leaves roughly the same number of uncovered pairs as the random triangle removal process. The paper presents this as a natural conjecture, motivated by the negligible impact of obstructions on more than four vertices during the early evolution; it remains open.