The limiting function conjecture for minimal H-factors

Let $H$ be a graph, let $d^*$ denote its maximal density parameter, and let $F_H(\alpha n,n)$ be the minimal weight of an $(\alpha,H)$-factor in the random weighted complete graph $K_n$. Here $alpha a[0,1]$ is the proportion of vertices not covered. Limiting function conjecture. There is a continuous decreasing function $f_H:[0,1]\to \mathbb{R}$ such that

$$F_H(\alpha n,n)/n^{1-1/d^*}\overset{\mathbb{P}}{\to} f_H(\alpha).$$

The preceding theorem gives the correct order of magnitude and sharp concentration for $F_H(\alpha n,n)$, but does not establish existence of this normalized limit; determining the function $f_H$ remains open.