Conjecture on the exact value of the weighted Hajnal–Szemerédi threshold

For integers $r\geq 2$ and $t\in(0,1]$, let $\delta(r,t)$ denote the infimum of the minimum weighted-degree density guaranteeing a partition into $r$-cliques of total weight at least $t\binom{r}{2}$. The authors' conjecture.

$$\delta(r,t)=\frac{1}{r}+\left(1-\frac{1}{r}\right)t.$$

The paper introduces this function and proves matching lower and upper bounds only in special or approximate forms, so the asserted exact formula remains open in the source.