Isaev–McKay–Southwell–Zhukovskii decomposition conjecture for random regular graphs

For integers $d_1,d_2\in[1,n-1]$ with $d_1\le d_2$, let $G(n,d_1)\oplus G(n,d_2-d_1)$ be the random $d_2$-regular graph obtained by sampling independent random regular graphs of degrees $d_1$ and $d_2-d_1$, conditioning them to be edge-disjoint, and taking their union.

Isaev–McKay–Southwell–Zhukovskii conjecture. If $d_2=\omega(1)$, then there exists a coupling of $G_{\oplus}\sim G(n,d_1)\oplus G(n,d_2-d_1)$ and $G\sim G(n,d_2)$ such that

$$\mathbb{P}(G_{\oplus}=G)=1-o(1).$$

Equivalently, the disjoint-union model should be asymptotically indistinguishable from the uniform random $d_2$-regular graph in a strong coupling sense. The source attributes this conjecture to Isaev, McKay, Southwell and Zhukovskii and presents it as an additional restriction related to sprinkling; its general validity remains open.