Injective coloring conjecture for regular uniform hypergraphs

Let $r\geq 2$. An injective $q$-coloring of an $r$-uniform hypergraph is a coloring of its vertices with $q$ colors such that any two edges sharing $r-1$ vertices have distinct color sets. Injective coloring conjecture. There exist constants $c,c'>0$ such that, whenever $H$ is a $\Delta$-regular $r$-graph with maximum $(r-1)$-degree at most $D$, $H$ has a spanning subgraph $G$ of minimum degree at least $c\Delta$ with an injective $c'D$-coloring. The source notes that an injectively colored subgraph with linearly many edges is already available; the open difficulty is preserving minimum degree, so the conjecture remains open.