pr.probability – Contiguity of uniform random regular graphs and uniform random regular graphs which have a perfect matching

Let us consider $cal{G}_{_{n,d}}$ as the uniform probability space of d-regular graphs
on the n vertices ${1, ldots, n }$ (where $dn$ is even). We say that an event $H_{_{n}}$ occurs a.a.s. (asymptotically almost surely) if $mathbf{P}_{_{cal{G}}}(H_{_{n}}) longrightarrow 1$ as $n ⟶ infty$.

Also, suppose $(cal{G}_{_{n}})_{_{n ≥ 1}}$ and $(cal{hat{G}}_{_{n}})_{_{n ≥ 1}}$ are two sequences of probability spaces such that $cal{G}_{_{n}}$ and $cal{hat{G}}_{_{n}}$ are differ only in the probabilities. We say that these sequences are contiguous if a sequence of events $A_{_{n}}$ is a.a.s. true in $cal{hat{G}}_{_{n}}$ if and only if it is true in $cal{hat{G}}_{_{n}}$, in which case we write
$$cal{G}_{_{n}} approx cal{hat{G}}_{_{n}}.$$

Thorem. (Bollobas) For any fixed $d geq 3$, $G_{_{n}} ∈ cal{G}_{_{n,d}}$ a.a.s has a perfect matching.

Using $cal{G}^p_{_{n,d}}$ to denote the uniform probability space of d-regular graphs which have a perfect matching on the n vertices ${1, ldots, n }$, is it true to conclude from the above theorem that $cal{G}_{_{n,d}} approx cal{G}^p_{_{n,d}}$?