Sandwich conjecture for random regular graphs

Let $N$ be the number of vertices, let $d=d(N)$, and let $\bm{G}_{N,d}$ denote a uniformly random $d$-regular graph. For edge probabilities $p_*$ and $p^*$, let $\bm{G}_*\sim\bm{G}_{N,p_*}$ and $\bm{G}^*\sim\bm{G}_{N,p^*}$ be Erdős–Rényi random graphs. Sandwich conjecture. If $d=d(N)$ dominates $\log N$ asymptotically, there exist

$$p_*=(1-o(1))\,d/N,\qquad p^*=(1+o(1))\,d/N,$$

such that

$$\mathbb{P}\bigl(\bm{G}_*\subseteq\bm{G}_{N,d}\subseteq\bm{G}^*\bigr)=1-o(1),$$

where $\subseteq$ denotes inclusion of edges. The conjecture compares uniformly random regular graphs with Erdős–Rényi graphs of asymptotically matching edge density; it is known in some degree ranges, but it remains open for the missing range $\log N\ll d\ll \log^4 N/\log^3\log N$.