Large Y3,2-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 $max\left(\left(\frac{4\alpha n}{3}\right),\left(\frac{n}{3}\right)-\left(\frac{n-\alpha n}{3}\right)\right)+o\left(n^3\right)$ 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 Erdős.