Sprinkling conjecture for random regular graphs

Let $d_1=d_1(n),d_2=d_2(n)\in\mathbb{N}:=\{0,1,\ldots\}$ satisfy

$$d_1+d_2\leq n-1,\qquad \min\{d_1+d_2,n-d_1,n-d_2\}\rightarrow\infty.$$

Let $\mathcal{S}_n(d_1,d_2)$ be the set of pairs of edge-disjoint graphs on $[n]$ whose respective degrees are $d_1$ and $d_2$, and let $(\boldsymbol{G}_1,\boldsymbol{G}_2)$ be uniform in this set. Write $\mathcal{G}(n,d)$ for the uniform random labelled $d$-regular graph. Sprinkling conjecture. There is a coupling $(\boldsymbol{G}_1,\boldsymbol{G}_2,\boldsymbol{G}_{d_1},\boldsymbol{G}_{d_2},\boldsymbol{G}_{d_1+d_2})$ such that

$$\mathbb{P}(\boldsymbol{G}_{d_1}=\boldsymbol{G}_1,\boldsymbol{G}_{d_2}=\boldsymbol{G}_2,\boldsymbol{G}_{d_1+d_2}=\boldsymbol{G}_1\cup\boldsymbol{G}_2)=1-o(1),$$

and $\boldsymbol{G}_{d_1}\sim\mathcal{G}(n,d_1)$, $\boldsymbol{G}_{d_2}\sim\mathcal{G}(n,d_2)$, and $\boldsymbol{G}_{d_1+d_2}\sim\mathcal{G}(n,d_1+d_2)$. This would establish that a random regular graph can asymptotically be sprinkled as two independent-looking edge-disjoint regular graphs; the conjecture is open in the stated generality.