Normality conjecture for spanning subgraph counts in random regular graphs

Let ${\mathcal G}(n,d)$ be the uniform random $d$-regular graph on $n$ vertices. A perfect matching is a spanning $1$-regular subgraph, a Hamilton cycle is a spanning cycle, a spanning tree is a spanning tree, and a $k$-factor is a spanning $k$-regular subgraph.

Normality conjecture. The numbers of perfect matchings, Hamilton cycles, spanning trees, and $k$-factors where $k\le d-1$, are all asymptotically normally distributed in ${\mathcal G}(n,d)$ for all $d$ where $dn$ is even and $\min\{d,n-d\}\to\infty$ as $n\to\infty$; here $n$ is even for perfect matchings and $kn$ is even for $k$-factors.

The conjecture extends the paper's normal-limit theorem for perfect matchings under the more restrictive condition $d=O(n^{1/7}/\log^2 n)$ and proposes a common limiting-distribution phenomenon for several spanning subgraph counts.