The minimum vertex-degree conjecture for spanning components in hypergraphs

Let $G$ be a $k$-graph on $n$ vertices. Write $\delta_1(G)$ for its minimum vertex degree, the minimum number of edges containing any one vertex. Spanning-component conjecture. If

$$\delta_1(G) > \tfrac{1}{2}\binom{n-1}{k-1},$$

then $G$ contains a spanning component. The statement is known for $k\leq 3$ and is conjectured for $k>3$; the threshold is asymptotically best possible.