Reiher–Rödl–Schacht conjecture on uniform Turán density and vanishing orders

Let $F$ be a $k$-graph. For integers $1\leq\ell\leq k-3$, let $\pi_{\rm u}^{(\ell)}(F)$ denote its $\ell$-uniform Turán density. An $(\ell+1)$-vanishing order is an ordering of $V(F)$ admitting an $(\ell+1)$-vanishing coloring in the sense that every $(\ell+1)$-subset of positions in each edge receives its corresponding color. Reiher–Rödl–Schacht conjecture. For $1\leq\ell\leq k-3$,

$$\pi_{\rm u}^{(\ell)}(F)=0\quad\Longleftrightarrow\quad F\text{ has an }(\ell+1)\text{-vanishing order}.$$

The conjecture connects zero uniform Turán density with the structural vanishing-order property; its resolution status is not specified in the supplied text.