The approximate packing conjecture for Hamilton cycles in random hypercube subgraphs

Let $\mathcal{Q}^n$ be the $n$-dimensional hypercube, let $\mathcal{Q}^n_p$ be its random subgraph obtained by retaining each edge independently with probability $p$, and let $\delta(\mathcal{Q}^n_p)$ denote its minimum degree. For every $p\in(1/2,1]$ and every $\eta>0$, the approximate Hamilton-cycle packing conjecture. Asymptotically almost surely, $\mathcal{Q}^n_p$ contains $(1/2-\eta)\delta(\mathcal{Q}^n_p)$ pairwise edge-disjoint Hamilton cycles. This is proposed as an approximate analogue in the hypercube of known packing results for random graphs; the supplied text gives no resolution.