Covering 3-uniform hypergraphs by vertex-disjoint tight paths

For alpha > 0 and large integer n, let H be an n-vertex 3-uniform hypergraph such that every pair of vertices is in at least n / 3 + alpha ( n ) $n\unicode{x02215}3+\alpha (n)$ edges. We show that H $H$ contains two vertex-disjoint tight paths whose union covers the vertex set of H $H$. The quantity two here is best possible and the degree condition is asymptotically best possible. This result also has an interpretation as the deficiency problems, recently introduced by Nenadov, Sudakov, and Wagner: every such H $H$ can be made Hamiltonian by adding at most two vertices and all triples intersecting them.