  For a $k$-uniform hypergraph (or simply $k$-graph) $F$, the codegree
Tur\'{a}n density $\pi_{\mathrm{co}}(F)$ is the supremum over all $\alpha$ such
that there exist arbitrarily large $n$-vertex $F$-free $k$-graphs $H$ in which
every $(k-1)$-subset of $V(H)$ is contained in at least $\alpha n$ edges.
Recently, it was proved that for every $3$-graph $F$, $\pi_{\mathrm{co}}(F)=0$
implies $\pi_{\therefore}(F)=0$, where $\pi_{\therefore}(F)$ is the uniform
Tur\'{a}n density of $F$ and is defined as the supremum over all $d$ such that
there are infinitely many $F$-free $k$-graphs $H$ satisfying that any induced
linear-size subhypergraph of $H$ has edge density at least $d$.
  In this paper, we introduce a layered structure for $3$-graphs which allows
us to obtain the reverse implication: every layered $3$-graph $F$ with
$\pi_{\therefore}(F)=0$ satisfies $\pi_{\mathrm{co}}(F)=0$. Along the way, we
answer in the negative a question of Falgas-Ravry, Pikhurko, Vaughan and Volec
[J. London Math. Soc., 2023] about whether
$\pi_{\therefore}(F)\leq\pi_{\mathrm{co}}(F)$ always holds. In particular, we
construct counterexamples $F$ with positive but arbitrarily small
$\pi_{\mathrm{co}}(F)$ while having $\pi_{\therefore}(F)\ge 4/27$.
