Alon–Wei asymptotic nearly-uniform degree-distribution conjecture

Let $G$ be a $d$-regular graph on $n$ vertices, let $H$ be a spanning subgraph of $G$, and suppose that $d=o(n)$. For each $k\in\{0,1,\dots,d\}$, write $\#\{v\in V(H):d_H(v)=k\}$ for the number of vertices of degree $k$ in $H$. Alon–Wei asymptotic conjecture. Every $d$-regular graph on $n$ vertices contains a spanning subgraph $H$ such that

$$\#\{v\in V(H):d_H(v)=k\}=(1+o(1))\frac{n}{d+1}$$

for all $0\leq k\leq d$. This is the asymptotic version proposed by Alon and Wei; the supplied abstract says that the paper proves it, so the conjecture is solved by the result described in the source.