Packing tetrahedrons in edge-weighted graphs

We prove that for all $\mu >0,t\in (0,1)$ and sufficiently large $n\in 4\mathbb{N}$, if $G$ is an edge-weighted complete graph on $n$ vertices with a weight function $w:E(G)\to [0,1]$ and the minimum weighted degree ${\delta }^w(G)\ge (\frac{1+3t}{4}+\mu )n$, then $G$ contains a $K_4$-factor where each copy of $K_4$ has total weight more than $6t$. This confirms a conjecture of Balogh–Kemkes–Lee–Young for the tetrahedron case.