The minimum codegree conjecture for spanning spheres in 3-graphs

Let $\mathcal{H}$ be a $3$-graph on $n>3$ vertices, and let $\delta_2(\mathcal{H})$ denote its minimum codegree.

Spanning-sphere conjecture. Every such $\mathcal{H}$ with

$$\delta_2(\mathcal{H})>n/3$$

contains a spanning copy of the sphere $\mathbb{S}^2$.

The conjecture would eliminate the error term in the paper's asymptotic minimum-codegree result for spanning spheres in $3$-graphs; its resolution is not given here.