Prescribed-parity perfect matching conjecture in random graphs

Let $v_1,\ldots,v_n$ be the vertices of $G(n,p)$, and let $\phi:[n]\to\{0,1\}$ satisfy

$$\left|\{i\in[n]:\phi(i)=1\}\right|\text{ is even}.$$

The prescribed parity class is the set of vertices whose degrees satisfy the congruences specified by $\phi$.

Prescribed-parity matching conjecture. If $p\geq n^{-1/2}$, then asymptotically almost surely $G(n,p)$ contains a perfect matching on

$$\left\{v_i:d_{G(n,p)}(v_i)\equiv \phi(i)-1\pmod 2\right\}.$$

This is a stronger statement than the odd-degree matching conjecture and is motivated by the need to realize the optimal leave in a final random slice. The source presents it as an open question.