The cycle-factor conjecture for random hypercube subgraphs

Let $\mathcal{Q}^n$ be the $n$-dimensional hypercube, and let $\mathcal{Q}^n_p$ be the random subgraph obtained by retaining each edge independently with probability $p$. For fixed $\varepsilon>0$ and integer $\ell\geq 2$, the cycle-factor conjecture. If $p\geq 1/2+\varepsilon$, then asymptotically almost surely $\mathcal{Q}^n_p$ contains a $C_{2^\ell}$-factor, namely a set of vertex-disjoint cycles of length $2^\ell$ whose union contains all vertices of $\mathcal{Q}^n$. The paper presents this as a further question related to embedding large subgraphs; no resolution is given in the supplied text.