On illumination of the boundary of a convex body in $\mathbb{E}^n$, $n=4,5,6$

Let $H_n$ be the minimal number of smaller homothetic copies of an $n$-dimensional convex body required to cover the whole body. Equivalently, $H_n$ can be defined via illumination of the boundary of a convex body by external light sources. The best known upper bound in three-dimensional case is $H_3\le 16$ and is due to Papadoperakis. We use Papadoperakis' approach to show that $H_4\le 96$, $H_5\le 1091$ and $H_6\le 15373$ which significantly improve the previously known upper bounds on $H_n$ in these dimensions.