Asymptotic equivalence conjecture for sums of random regular graphs

For $d_1,d_2\in\mathbb{N}$ with $d_1+d_2\leq n-1$, let $\mathcal{G}(n,d)$ denote the uniform random $d$-regular graph. Let $\mathcal{G}(n,d_1)\oplus\mathcal{G}(n,d_2)$ denote the distribution obtained by sampling independent random regular graphs of degrees $d_1$ and $d_2$, conditioned to be edge-disjoint. Asymptotic equivalence conjecture. If $d_1+d_2\rightarrow\infty$, there is a coupling $(\boldsymbol{G}_{\oplus},\boldsymbol{G}_{d_1+d_2})$ such that

$$\boldsymbol{G}_{\oplus}\sim\mathcal{G}(n,d_1)\oplus\mathcal{G}(n,d_2),\qquad \boldsymbol{G}_{d_1+d_2}\sim\mathcal{G}(n,d_1+d_2),$$

and

$$\mathbb{P}(\boldsymbol{G}_{\oplus}=\boldsymbol{G}_{d_1+d_2})=1-o(1).$$

For fixed positive degrees, only contiguity was previously known in the stated nonexceptional cases; the stronger coupling assertion remains open.