Finding long cycles in a percolated expander graphs

Given a graph $G$, the percolated graph $G_p$ has each edge independently retained with probability $p$. Collares, Diskin, Erde, and Krivelevich initiated the study of large structures in percolated single-scale vertex expander graphs, wherein every set of exactly $k$ vertices of $G$ has at least $dk$ neighbours before percolation. We extend their result to a conjectured stronger form, proving that if $p=(1+\epsilon )\mathrm{/}d$ and $G$ is a graph on at least $k$ vertices which expands as above, then $G_p$ contains a cycle of length ${\mathrm{\Omega }}_{\epsilon }(kd)$ with probability at least $1-\exp (-{\mathrm{\Omega }}_{\epsilon }(k\mathrm{/}d))$ as $k\to \mathrm{\infty }$.