A minimum-degree threshold on 2^n vertices forcing the n-cube Q_n (Erdős #1035)

The $n$-dimensional hypercube $Q_n$ is the graph on the $2^n$ binary strings of length $n$, two strings adjacent exactly when they differ in a single coordinate. Question: is there an absolute constant $c>0$ such that, for every $n$, every graph on $2^n$ vertices with minimum degree greater than $(1-c)2^n$ contains $Q_n$ as a subgraph?

Acceptance. FULLY RESOLVES: prove that such a constant $c>0$ exists — give an explicit $c$ and a proof that minimum degree $>(1-c)2^n$ forces $Q_n$ for all $n$ — OR disprove it by constructing, for $c\to 0$, graphs on $2^n$ vertices with minimum degree $>(1-c)2^n$ that contain no $Q_n$. Complete proof (Lean/Coq preferred, otherwise a full written proof). ADVANCES, any of: (a) for a fixed small $n$, determine the exact minimum-degree threshold on $2^n$ vertices that forces $Q_n$ (the value $u_n$ of question (ii)), delivering an extremal $Q_n$-free graph plus an exhaustiveness certificate that any larger minimum degree forces $Q_n$; (b) establish the first nontrivial value of $c$ valid for an infinite family of $n$, with proof; (c) prove nontrivial bounds on the smallest $m$ of question (i). Deliver the proof, or the extremal graph(s) + threshold/exhaustiveness certificate.