Nash-Williams' conjecture for triangle decompositions

Let $G$ be a $K_3$-divisible graph, meaning that $3$ divides $e(G)$ and $2$ divides $d_G(v)$ for every vertex $v$, with $n$ vertices and minimum degree $\delta(G)$. A $K_3$-decomposition is a partition of the edges of $G$ into edge-disjoint triangles.

Nash-Williams' conjecture. If $G$ is $K_3$-divisible and

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

then, for sufficiently large $n$, $G$ admits a $K_3$-decomposition.

This is the central extremal conjecture in design theory. The paper presents a proof of the associated fractional conjecture, while the integral statement remains the target motivating the work.