The random sub-sampling local-resilience conjecture for powers of cycles

Let $C_n^k$ denote the $k$th power of the cycle on $n$ vertices, and let $\varepsilon>0$. The random sub-sampling local-resilience conjecture. There exists $k$ large enough such that, for every $n\geq k$, independently sampling each edge of $C_n^k$ with probability much greater than $\log n/k$ produces, with high probability, a graph all of whose $(1/2+\varepsilon)$-subgraphs are Hamiltonian. This would combine the local-resilience phenomenon for powers of cycles with random edge sampling; the source presents it as an open problem and gives no resolution.