Characterization of zero uniform Turán density for r-graphs

Let $F$ be an $r$-graph. An ordering $\preceq$ on $V(F)$ and a function

$$\varphi:\binom{V(F)}{2}\rightarrow\binom{[r]}{2}$$

assign to each pair of vertices the role it plays in an edge containing that pair. The zero-density characterization conjecture.

$$\pi_{\!\text{uniform}}(F)=0$$

if and only if there exists an ordering $\preceq$ on $V(F)$ and such a function $\varphi$ with the property that, for each pair of vertices $u,v$ and every edge $e$ containing $u$ and $v$, the pair $\{u,v\}$ plays the role $\varphi(uv)$ in $e$.

For $3$-graphs, Reiher, Rödl and Schacht proved the analogous characterization. Extending it to $r$-graphs for general $r$ would characterize exactly the forbidden hypergraphs of zero uniform Turán density; the general case is left open in the source.