Expected regular-subgraph count conjecture for random regular graphs

Let $\boldsymbol{G}_{d_1+d_2}\sim\mathcal{G}(n,d_1+d_2)$, and let $\mathcal{R}_{d_1}(G)$ be the set of $d_1$-regular spanning subgraphs of $G$. For $d_1,d_2>0$ with $d_1+d_2\leq n-1$, define $\lambda:=d_1/(d_1+d_2)$. Expected regular-subgraph count conjecture.

$$\mathbb{E}|\mathcal{R}_{d_1}(\boldsymbol{G}_{d_1+d_2})|=(2^{1/2}e^{1/4}+o(1))\bigl(\lambda^\lambda(1-\lambda)^{1-\lambda}\bigr)^{n(d_1+d_2)/2}\binom{d_1+d_2}{d_1}^{n}.$$

This conjecture generalises the known complete-graph enumeration formula and is proved in the paper when $d_1,d_2=\omega(n/\log n)$; it remains open in general.