Higher-degree vanishing-order conjecture

For integers $k\geq4$ and $\ell\geq3$ with $\ell<k$, let $F$ be a $k$-graph. An $\ell$-vanishing order of $F$ is an ordering of $V(F)$ admitting an $\ell$-vanishing coloring with the property that, on every edge, each $\ell$-subset of positions receives its corresponding color. Higher-degree vanishing-order conjecture. If

$$\pi_\ell(F)=0,$$

then $F$ has an $\ell$-vanishing order. The proved $\ell=2$ case motivates this proposed extension to all $\ell\geq3$.