The exact extremal number of $C_4$-free subgraphs of $Q_8$

Let $Q_8$ be the 8-dimensional hypercube and let $C_4$ denote the cycle of length four. Write $\mathrm{ex}(Q_8,C_4)$ for the maximum number of edges in a subgraph of $Q_8$ containing no copy of $C_4$. The $Q_8$ extremal conjecture.

$$\mathrm{ex}(Q_8,C_4)=680.$$

A 680-edge $C_4$-free subgraph is constructed and certified, and every non-edge creates a $C_4$ when added. The equality is not established because the reported computational search for 681 edges found no $C_4$-free example.