Balogh, Kostochka and Treglown's conjecture on degree sequences forcing perfect clique tilings

Let $n,r\in\mathbb{N}$ with $r$ dividing $n$. Suppose that $G$ is a graph on $n$ vertices with degree sequence $d_1\leq\ldots\leq d_n$ satisfying the following conditions:

$$d_i\geq (r-2)n/r+i\quad\text{for all }i<n/r;$$ $$d_{n/r+1}\geq (r-1)n/r.$$

Balogh, Kostochka and Treglown's conjecture. Then $G$ contains a perfect $K_r$-tiling, that is, a collection of vertex-disjoint copies of $K_r$ covering all vertices of $G$.

This conjecture seeks a degree-sequence condition forcing a perfect $H$-tiling that is best possible in a weaker sense than a Chvátal-type condition. It concerns the degree-sequence analogue of perfect clique-tiling results and remains open in the supplied source context.