High-girth Nash-Williams conjecture for triangle decompositions

Let $G$ be a graph on $|G|$ vertices. It is $K_3$-divisible if its number of edges is divisible by $3$ and every vertex degree is even. A triangle decomposition with girth at least $g$ is a partition of the edges into triangles such that no set of fewer than $g$ vertices spans at least two fewer triangles than vertices.

High-girth Nash-Williams conjecture. For every fixed $g$, every sufficiently large $K_3$-divisible graph $G$ with

$$\delta(G)\ge 3|G|/4$$

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

This combines the minimum-degree triangle-decomposition conjecture with Erdős's local-sparsity requirement. The source presents it as a proposed combination of two unresolved conjectures.