Exact minimum codegree threshold for $K_4^{-}$-factors

Given hypergraphs $F$ and $H$, an $F$-factor in $H$ is a set of vertex-disjoint copies of $F$ which cover all the vertices in $H$. Let $K_4^{-}$ denote the $3$-uniform hypergraph with $4$ vertices and $3$ edges. We show that for sufficiently large $n\in 4\mathbb{N}$, every $3$-uniform hypergraph $H$ on $n$ vertices with minimum codegree at least $n\mathrm{/}2-1$ contains a $K_4^{-}$-factor. Our bound on the minimum codegree here is best-possible. It resolves a conjecture of Lo and Markstr"om for large hypergraphs, who earlier proved an asymptotically exact version of this result. Our proof makes use of the absorbing method as well as a result of Keevash and Mycroft concerning almost perfect matchings in hypergraphs.