Vanishing codegree Turán density implies vanishing uniform Turán density

For a $k$-uniform hypergraph (or simply $k$-graph) $F$, the codegree Turán density ${\pi }_{\mathrm{c}\mathrm{o}}(F)$ 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 Turán 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 $F$, ${\pi }_{\therefore }(F)\le {\pi }_{\mathrm{c}\mathrm{o}}(F)$. We provide a positive answer to this question provided that ${\pi }_{\mathrm{c}\mathrm{o}}(F)=0$. Our proof relies on a random geometric construction and a new formulation of the characterization of $3$-graphs with vanishing uniform Turán density due to Reiher, R{ö}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.