Illingworth–Lang–Müyesser–Parczyk–Sgueglia's spanning tight component conjecture

Let $H$ be an $n$-vertex $k$-uniform hypergraph. Its minimum codegree is the minimum, over all $(k-1)$-tuples $S\in\binom{V(H)}{k-1}$, of the number of edges containing $S$. The hypergraph is tightly connected if, for any $e,e'\in E(H)$, there is a sequence of edges $f_0,\ldots,f_\ell\in E(H)$ with $e=f_0$, $e'=f_\ell$, and $|f_i\cap f_{i+1}|=k-1$ for every $i\in[\ell-1]$. A tight component is an edge-maximal tightly connected subgraph, and it is spanning if every vertex of $H$ lies in one of its edges.

Illingworth–Lang–Müyesser–Parczyk–Sgueglia's conjecture. If $H$ has minimum codegree at least $n/k$, then $H$ has a spanning tight component.

This conjecture identifies $n/k$ as the conjectural threshold for global tight connectivity in uniform hypergraphs. Extremal constructions show that the threshold cannot generally be lowered to around $n/k$, while the asserted existence at the threshold remains open.