Odd-vertex perfect matching conjecture in random graphs

Let $G(n,p)$ be the binomial random graph. Its odd-degree vertices are the vertices $v$ for which $d_{G(n,p)}(v)$ is odd.

Odd-vertex matching conjecture. If

$$p\geq n^{-1/2},$$

then asymptotically almost surely $G(n,p)$ contains a perfect matching on its odd-degree vertices.

This would attain the degree-parity contribution to the optimal triangle-packing leave. The source notes that the claim may hold already for $p\gg n^{-1}\log n$, but only states the displayed range and leaves it as an open question.