Gao–Isaev–McKay embedding conjecture for random regular graphs

Let $\mathcal{G}(n,d)$ denote the uniform random labelled $d$-regular graph. Let $0\leq d_1\leq d\leq n-1$ be integers, excluding $(d_1,d)=(1,2)$ and $(d_1,d)=(n-3,n-2)$. Gao–Isaev–McKay embedding conjecture. There exists a coupling $(\boldsymbol{G}_{d_1},\boldsymbol{G}_d)$ such that $\boldsymbol{G}_{d_1}\sim\mathcal{G}(n,d_1)$, $\boldsymbol{G}_d\sim\mathcal{G}(n,d)$, and

$$\mathbb{P}(\boldsymbol{G}_{d_1}\subseteq\boldsymbol{G}_d)=1-o(1).$$

The conjecture is known in several ranges of degrees, including the regimes listed in the paper, but remains open in full generality.