Alon's Question on Connectivity Graph-Codes: $f(d)=2^d$ for Every $d\geq 4$

For a finite graph $H$, a connectivity graph-code is a family $\mathcal C\subseteq 2^{E(H)}$ such that $A\triangle B$ is a connected spanning subgraph of $H$ whenever $A$ and $B$ are distinct members of $\mathcal C$. Let $m(H)$ denote the maximum size of such a family, and let $f(d)$ be the largest