Erdős meets Nash-Williams' conjecture for high-girth triangle decompositions

Let $G$ be a $K_3$-divisible graph, meaning that $3$ divides $e(G)$ and every vertex has even degree. The girth of a $K_3$-decomposition is the smallest integer $g$ for which some $g$ triangles span at most $g+2$ vertices.

Erdős meets Nash-Williams' conjecture. For every integer $g$, every sufficiently large $K_3$-divisible graph $G$ on $n$ vertices satisfying

$$\delta(G)\geq \frac{3n}{4}$$

admits a $K_3$-decomposition with girth at least $g$.

This conjecture combines the high-girth Steiner triple system problem with Nash-Williams' minimum-degree problem. The source presents it as a conjectural common generalization; no resolution is supplied for the exact $3n/4$ threshold.