Two-state component-size conjecture for 1-independent percolation on the hypercube

Let $Q_n$ be the $n$-dimensional hypercube, let $p\in (\frac{1}{2},1]$ be fixed, and let $\mathcal{M}_{1,\geq p}(Q_n)$ denote the $1$-independent percolation measures on $Q_n$ whose edge marginals are at least $p$. For a measure $\mu$, write $\mathbf{H}_{\mu}$ for the resulting random subgraph and $C_1(\mathbf{H}_{\mu})$ for its largest connected component. Two-state component-size conjecture. For all $\mu\in\mathcal{M}_{1,\geq p}(Q_n)$, with probability $1-o(1)$,

$$\left\vert C_1\left(\mathbf{H}_{\mu}\right)\right\vert \geq \left(\frac{1+\sqrt{2p-1}}{2}-o(1)\right)2^n.$$

The conjecture says that the two-state measure asymptotically minimises the size of the largest component, extending the paper's exact finite-graph result and its corresponding asymptotic theorem for weakly pseudorandom graphs. It remains open for the hypercube.