Large $ Y_{3,2} $-tilings in 3-uniform hypergraphs

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.