The codegree conjecture for spanning spheres in uniform hypergraphs

Let $rgeq 2$, and let $\mathcal{H}$ be an $r$-graph on $n>r$ vertices. For a set of $r-1$ vertices, its codegree is the number of edges containing it, and a tight component is an equivalence class under the transitive closure of the relation that two edges satisfy when they meet in $r-1$ vertices.

Higher-dimensional spanning-sphere conjecture. If $\mathcal{H}$ comprises a single tight component and every set $Z$ of $r-1$ vertices contained in an edge of $\mathcal{H}$ is contained in at least $n/2$ edges, then $\mathcal{H}$ contains a spanning copy of the $(r-1)$-dimensional sphere $\mathbb{S}^{r-1}$.

This conjecture is motivated by a construction showing that a spanning tight component alone does not force a spanning surface. It predicts a codegree threshold of $1/2$, independent of the dimension, but no resolution is supplied.