On-line Packing Cubes into $n$ Unit Cubes

If $n\geq 3$ and $d\in \{3,4\}$ or if $n\geq 1$ and $d\geq 5$, then any sequence of $d$-dimensional cubes of edge lengths not greater than $1$ whose total volume does not exceed $(n+1)\cdot 2^{-d}$ can be on-line packed into $n$ unit $d$-dimensional cubes.