ELW's Hamiltonicity conjecture for subgraphs of powers of cycles

Let $C_n^k$ be the $k$-th power of the cycle on $n$ vertices, and let $\delta(G)$ denote the minimum degree of a graph $G$. ELW's Hamiltonicity conjecture. For all integers $n\geq3$ and $k\in [1,n/2]$, every graph $G\subseteq C_n^k$ with $\delta(G)\geq k+1$ is Hamiltonian. This conjecture concerns a sufficient minimum-degree condition for subgraphs of powers of cycles to contain a Hamilton cycle and arises from the study of local resilience in random geometric graphs; its resolution is not given here.