Contiguity conjecture for unions of random perfect matchings

Let $n$ be even and let $d\in[3,n-3]$. For each even $n$, fix a set $A_n$ of $d$-regular graphs on $n$ vertices. Let $G(n,1)\oplus\dotsb\oplus G(n,1)$ denote the disjoint union of $d$ random perfect matchings.

Random matching decomposition conjecture. The event that the disjoint union belongs to $A_n$ holds with high probability if and only if the event that a uniform random $d$-regular graph belongs to $A_n$ holds with high probability:

$$G(n,1)\oplus\dotsb\oplus G(n,1)\in A_n\text{ with high probability}\quad\Longleftrightarrow\quad G(n,d)\in A_n\text{ with high probability}.$$

The paper proves one implication for $d\le n^{1/10}$ and states that, more generally, both the implication and its converse are believed to hold throughout $d\in[3,n-3]$.