Erdős–Faudree conjecture on Hamiltonian subsets of regular Dirac graphs

Let $G$ be an $(n+1)$-regular graph on $2n$ vertices. A subset $S\subseteq V(G)$ is called Hamiltonian when the induced subgraph $G[S]$ contains a Hamilton cycle. Erdős–Faudree conjecture. There is an absolute constant $c>0$ such that $G$ has at least $c2^{2n}$ Hamiltonian subsets. The source presents this as a question of Erdős and Faudree; no resolution is supplied in the given text.