Apex extension conjecture for degree Turán densities

Let $F$ be a $(k-1)$-graph, and let $\mathcal{S}_F$ be the $k$-graph obtained by adjoining an apex vertex $v$ and including $v\cup e$ as an edge for every $e\in E(F)$. For $2\leq\ell<k$, one always has $\pi_\ell(\mathcal{S}_F)\leq\pi_{\ell-1}(F)$. Apex extension conjecture. For integers $3\leq\ell<k$,

$$\pi_\ell(\mathcal{S}_F)=0\quad\Longleftrightarrow\quad\pi_{\ell-1}(F)=0.$$

The forward implication is known for $\ell=2$ via the paper's $2$-vanishing-order theorem, while the analogous equivalence for $\ell\geq3$ is proposed as an open direction.