Balogh–Kemkes–Lee–Young's weighted Hajnal–Szemerédi conjecture

Let $G=(V,E,w)$ be an edge-weighted complete graph on $n$ vertices, where $w:E\to[0,1]$. For $v\in V$, define its weighted degree by $d^w(v):=\sum_{uv\in E}w(uv)$ and let $\delta^w(G):=\min\{d^w(v):v\in V\}$. A copy of $K_r$ is $t$-heavy if the sum of the weights of its $\binom{r}{2}$ edges is strictly greater than $\binom{r}{2}t$; a $K_r$-factor is a collection of vertex-disjoint copies of $K_r$ covering $V$. Balogh–Kemkes–Lee–Young's conjecture. For any $\mu>0$, $r\ge 2$, $t\in(0,1)$, and any sufficiently large $n\in r\mathbb{N}$, if

$$\delta^w(G)\ge \left(\frac{1}{r}+\left(1-\frac{1}{r}\right)t+\mu\right)n,$$

then $G$ contains a $t$-heavy $K_r$-factor. This is a weighted analogue of the Hajnal–Szemerédi theorem, asking for the minimum weighted-degree threshold guaranteeing a perfect heavy clique tiling; the supplied parser gives no evidence that the conjecture has been resolved.