The degree-sequence conjecture for perfect clique packings

Let $n,r\in\mathbb N$ with $r$ dividing $n$. Let $G$ be a graph on $n$ vertices with degree sequence $d_1\leq\dots\leq d_n$. A perfect $K_r$-packing is a collection of vertex-disjoint copies of $K_r$ covering all vertices of $G$.

Degree-sequence conjecture. If

$$d_i\geq (r-2)n/r+i\quad\text{for all }i<n/r,$$

and

$$d_{n/r+1}\geq (r-1)n/r,$$

then $G$ contains a perfect $K_r$-packing.

This extends the Hajnal–Szemerédi theorem because the condition permits $n/r$ vertices to have degree below $(r-1)n/r$. The degree condition is essentially best possible, and the conjecture is proved when $G$ is additionally $K_{r+1}$-free; the general case remains open.