Layered 3-graph characterization of vanishing codegree Turán density

A $3$-graph $F$ is layered if there is a function $f:V(F)\to\mathbb N$ such that: every edge has exactly one vertex whose label is strictly greater than the other two; edges with the same maximum label have the same multiset of labels; and whenever edges $uvw$ and $u'v'w'$ satisfy $f(u)=f(u')$ and $f(v)=f(v')$, they also satisfy $f(w)=f(w')$. Let $\pi_{\mathrm{co}}(F)$ denote the codegree Turán density of $F$, and let $\pi_{\mathrm{u}}(F)$ denote its uniform Turán density. Layered 3-graph conjecture. For every $3$-graph $F$,

$$\pi_{\mathrm{co}}(F)=0 \quad\Longleftrightarrow\quad F\text{ is layered and }\pi_{\mathrm{u}}(F)=0.$$

The paper proves this equivalence for layered $3$-graphs, so the conjecture asserts that the layered condition is also necessary. The supplied text gives no resolution of the full characterization.