Maximum diameter of redundant non-surrounded graphs

Let $G$ be a graph of order $n$ in which every vertex is redundant and no vertex is surrounded. Its diameter is the graph distance between vertices at maximum distance.

Maximum-diameter conjecture. The maximum diameter of such a graph is

$$\left\lfloor \log_2(n)\right\rfloor;$$

when $n$ is a power of $2$, this maximum is attained only by the $k$-cube and by graphs containing the $k$-cube as a subgraph.

The $k$-cube is given in the surrounding text as an example with $2^k$ vertices, diameter $k$, all vertices redundant, and no surrounded vertices. The supplied text does not establish the asserted upper bound or the uniqueness claim.