Supercritical Site Percolation on the Hypercube: Small Components are Small

We consider supercritical site percolation on the $d$-dimensional hypercube $Q^d$. We show that typically all components in the percolated hypercube, besides the giant, are of size $O(d)$. This resolves a conjecture of Bollob'as, Kohayakawa, and {\L}uczak from 1994.