Optimal sparse blow-up conjecture for bounded-bandwidth graphs

For each $\gamma>0$, $\Delta\ge2$, and $k\ge1$, there should exist constants $\beta^\ast>0$ and $C^\ast>0$ such that, asymptotically almost surely for $\Gamma=G(n,p)$ whenever $p\ge C^\ast n^{-2/(\Delta+2)}$, every spanning subgraph $G$ of $\Gamma$ with

$$\delta(G)\ge\left(\frac{k-1}{k}+\gamma\right)pn$$

and every $k$-colourable graph $H$ on $n$ vertices with $\Delta(H)\le\Delta$, bandwidth at most $\beta^\ast n$, at least $C^\ast p^{-2}$ vertices not contained in any triangles, and at least $C^\ast p^{-(\Delta+2)(\Delta-1)/2}n^{2-\Delta}$ vertices not contained in any $K_{\Delta+1}$, satisfy the following. Sparse blow-up conjecture. The graph $G$ contains a copy of $H$. This conjecture proposes the expected optimal lower bound on $p$, accounting for the necessary obstructions caused by vertices of $H$ outside triangles and copies of $K_{\Delta+1}$.