Hasheminezhad–McKay conjecture on regular partitions of the complete graph

Let $d_0,d_1,\ldots,d_k\geq1$ satisfy

$$\sum_{i=0}^k d_i=n-1,$$

and let $R(n;d_0,\ldots,d_k)$ be the number of partitions of the edges of $K_n$ into spanning regular subgraphs of degrees $d_0,\ldots,d_k$. Define $\lambda_i:=d_i/(n-1)$. Hasheminezhad–McKay conjecture. If $k=o(n)$, then

$$R(n;d_0,\ldots,d_k)=(1+o(1))e^{k/4}2^{k/2}\mathopen{}\mathclose\bgroup\originalleft(\prod_{i=0}^k\lambda_i^{\lambda_i}\aftergroup\egroup\originalright)^{\binom{n}{2}}\binom{n-1}{d_0,\ldots,d_k}^{n}.$$

The expected regular-subgraph count conjecture above is a special case. The general partition-enumeration assertion is not resolved in the source.