The degree-sequence conjecture for perfect graph packings

Let $\gamma>0$ and let $H$ be a graph with chromatic number $\chi(H)=r$. A perfect $H$-packing is a collection of vertex-disjoint copies of $H$ covering all vertices. Let $G$ be a graph of order $n$ with degree sequence $d_1\leq\dots\leq d_n$.

Perfect-packing conjecture. There is an integer $n_0=n_0(\gamma,H)$ such that, whenever $n\geq n_0$, $|H|$ divides $n$, and

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

$G$ contains a perfect $H$-packing.

This is proposed as a likely consequence or application of the preceding perfect $K_r$-packing conjecture. No proof or resolution is given in the source, so the asymptotic generalization remains open.