  For a $k$-uniform hypergraph (or simply $k$-graph) $F$, the codegree
Tur\'{a}n density $\pi_{\mathrm{co}}(F)$ is the infimum over all $\alpha$ such
that any $n$-vertex $k$-graph $H$ with every $(k-1)$-subset of $V(H)$ contained
in at least $\alpha n$ edges has a copy of $F$. The uniform Tur\'{a}n density
$\pi_{\therefore}(F)$ is the supremum over all $d$ such that there are
infinitely many $F$-free $k$-graphs $H$ satisfying that any linear-size
subhypergraph of $H$ has edge density at least $d$. Falgas-Ravry, Pikhurko,
Vaughan and Volec [J. London Math. Soc., 2023] asked whether for every
$3$-graph $F$, $\pi_{\therefore}(F)\leq\pi_{\mathrm{co}}(F)$. We provide a
positive answer to this question provided that $\pi_{\mathrm{co}}(F)=0$. Our
proof relies on a random geometric construction and a new formulation of the
characterization of $3$-graphs with vanishing uniform Tur\'{a}n density due to
Reiher, R{\"o}dl and Schacht [J. London Math. Soc., 2018]. Along the way, we
answer a question of Falgas-Ravry, Pikhurko, Vaughan and Volec about
subhypergraphs with linear minimum codegree in uniformly dense hypergraphs in
the negative.
