The tetrahedron-component conjecture for dense 3-graphs

Let $G$ be a $3$-graph on $n$ vertices, and let $\delta_2(G)$ denote its minimum codegree. Define the $4$-graph $H$ on $V(G)$ by making a $4$-set an edge whenever it spans a tetrahedron in $G$. Tetrahedron-component conjecture. If

$$\delta_2(G) \geq 3n/4,$$

then $H$ is connected. This conjecture concerns the connectivity of tetrahedra in a dense $3$-graph and would constitute significant progress toward a $3$-uniform version of Pósa's conjecture on squares of cycles.