Kim–Vu sandwich conjecture for random regular graphs

Let $d$ be a degree parameter satisfying $d\gg\log n$. For sequences $p_1=p_1(n)\sim d/n$ and $p_2=p_2(n)\sim d/n$, consider a random $d$-regular graph $\mathbb R(n,d)$ and binomial random graphs $\mathbb G(n,p_1)$ and $\mathbb G(n,p_2)$. Kim–Vu sandwich conjecture. There is a joint distribution of these graphs such that, with high probability,

$$\mathbb G(n,p_1)\subseteq\mathbb R(n,d)\subseteq\mathbb G(n,p_2).$$

The conjecture seeks to couple a random regular graph between two binomial random graphs with asymptotically matching edge probabilities. Such a sandwiching result would provide a different route to proving that the paper's Bi-uniform construction yields a suitable point set; the source does not state whether the conjecture has been resolved.