The exact extremal number of $C_4$-free subgraphs of $Q_7$

Let $Q_7$ be the 7-dimensional hypercube and let $C_4$ denote the cycle of length four. Write $\mathrm{ex}(Q_7,C_4)$ for the maximum number of edges in a subgraph of $Q_7$ containing no copy of $C_4$. The $Q_7$ extremal conjecture.

$$\mathrm{ex}(Q_7,C_4)=304.$$

A 304-edge $C_4$-free subgraph is constructed and certified, with extensive computational evidence from independent searches that no 305-edge example was found. The exact upper bound remains unproved in the supplied text.