Subgraphs with large minimum $\mathrm{\ell }$-degree in hypergraphs where almost all $\mathrm{\ell }$-degrees are large

Let $G$ be an $r$-uniform hypergraph on $n$ vertices such that all but at most $\epsilon \left(\genfrac{}{}{0px}{}{n}{\mathrm{\ell }}\right)$ $\mathrm{\ell }$-subsets of vertices have degree at least $p\left(\genfrac{}{}{0px}{}{n-\mathrm{\ell }}{r-\mathrm{\ell }}\right)$. We show that $G$ contains a large subgraph with high minimum $\mathrm{\ell }$-degree.