The minimum codegree threshold conjecture for two tight paths in 3-graphs

Let $H$ be an $n$-vertex $3$-graph, and let $[?]$ denote its minimum codegree, the minimum number of common neighbours over all pairs of vertices. A collection of tight paths is vertex-disjoint if their vertex sets are pairwise disjoint, and it covers $V(H)$ if the union of their vertex sets is $V(H)$. Minimum codegree threshold conjecture. The minimum codegree assumption in the theorem guaranteeing two vertex-disjoint tight paths whose union covers $V(H)$ can be weakened to $n/3$; equivalently, minimum codegree at least $n/3$ should guarantee such a cover. This would make $n/3$ the asymptotically sharp threshold, since the stated three-part construction requires three tight paths.