Exact high-girth triangle-decomposition conjecture

Let $K_3$-divisibility mean that the number of edges is divisible by $3$ and every vertex degree is even. A triangle decomposition of a graph partitions its edges into copies of $K_3$, and its girth is the smallest size of a vertex set spanning at least two fewer triangles than vertices.

High-girth triangle-decomposition conjecture. For every fixed $g$, every sufficiently large $K_3$-divisible complete graph $K_n$ has a $K_3$-decomposition with girth at least $g$.

This is the quantified exact form of the high-girth Steiner triple system conjecture. The source reports that only the first nontrivial case is known exactly, while approximate results are available for all fixed $g$.