Gustavsson–Nash-Williams conjecture on clique decompositions

Let $r\ge 3$. A graph $G$ is $K_r$-divisible when its number of edges is divisible by $\binom{r}{2}$ and every vertex degree is divisible by $r-1$.

Gustavsson–Nash-Williams conjecture. For every $r\ge 3$, there exists an $n_0=n_0(r)$ such that every $K_r$-divisible graph $G$ on $n\ge n_0$ vertices with

$$\delta(G)\ge (1-1/(r+1))n$$

\nhas a $K_r$-decomposition.

This conjecture predicts the asymptotically sharp minimum-degree threshold for decomposing divisible graphs into copies of a clique. It is attributed to Nash-Williams for triangles and to Gustavsson in general; the supplied source gives no resolution status.