The $n$-dimensional hypercube $Q_n$ is the graph whose vertex set is $\{0, 1\}^n$ and whose edge set is the set of pairs that differ in exactly one coordinate. A graph is called cubical if it is a subgraph of $Q_n$ for some $n$.
We know that $|V(Q_n)|=2^n$ and $|E(Q_n)|=n 2^{n-1}$, so $|E(Q_n)|=\frac{1}{2}|V(Q_n)|\log_2 |V(Q_n)|$. Graham [On primitive graphs and optimal vertex assignments. Ann. New York Acad. Sci. 175 (1970), 170--186] showed that a $t$-vertex cubical graph can have at most $(1+o(1))\frac{1}{2}t\log_2 t$ edges.
My question is: can a $2^{n}$-vertex cubical graph have more than $n2^{n-1}$ edges? In general, for $2^{n-1}< t \leq 2^{n}$, can a $t$-vertex cubical graph have more edges than the subgraph of $Q_n$ induced by any $t$ vertices?