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{4αn}{3}, \binom{n}{3}-\binom{n-αn}{3} \right \}+o(n^3) $ edges contains a $Y_{3,2}$-tiling covering more than $ 4αn$ vertices, for sufficiently large $ n $ and $0<α< 1/4$. 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 Erdős.