Hamiltonicity conjecture for subgraphs of powers of paths

Let $P_n^k$ be the $k$-th power of the path on $n$ vertices, and let $\deg_G(v)$ denote the degree of $v$ in $G$. Hamiltonicity conjecture for path powers. Let $n\geq4$ and $k\geq2$ be integers. If $G\subseteq P_n^k$ satisfies

$$\deg_G(v)\geq \deg_{P_n^k}(v)/2+2$$

for every vertex $v$, then $G$ is Hamiltonian. Motivated by the paper's result for effective minimum degree, this conjecture asks whether preserving slightly more than half the degree of every vertex of a path power guarantees a Hamilton cycle; it would imply the one-dimensional case of the cited random-geometric-graph conjecture and remains open in the source.