Subdivision conjecture for percolated vertex expanders

Fix $d\in\mathbb{N}$ and let $H$ be a graph of maximum degree $3$. Let $k\in\mathbb{N}$, and let $G$ be an $(=\hspace{-0.2em}k,d)$-vertex expander, meaning that every set $S\subseteq V(G)$ with $|S|=k$ has at least $dk$ external neighbors. For $\varepsilon\geq\varepsilon_0(d)=d^{-O(1)}$, set $p=(1+\varepsilon)/d$ and let $G_p$ be the percolated graph. Subdivision conjecture. With high probability, $G_p$ contains a subdivision of $H$. This is proposed as a significant strengthening of the paper's main long-cycle theorem; the precise dependence of the threshold $\varepsilon_0(d)$ is part of the conjecture.