  Let $Y_{3,2}$ be the 3-graph with two edges intersecting in two vertices. We
prove that every 3-graph $ H $ on $ n $ vertices with at least $ \max \left \{
\binom{4\alpha n}{3}, \binom{n}{3}-\binom{n-\alpha n}{3} \right \}+o(n^3) $
edges contains a $Y_{3,2}$-tiling covering more than $ 4\alpha n$ vertices, for
sufficiently large $ n $ and $0<\alpha< 1/4$. The bound on the number of edges
is asymptotically best possible and solves a conjecture of the authors for
3-graphs that generalizes the Matching Conjecture of Erd\H{o}s.
