A non-lattice periodic point set beating the optimal lattice packing-covering constant in dimension five

The packing-covering constant of a point set $X\subseteq\mathbb{R}^d$ is $γ(X)=μ(X)/ρ(X)$, the covering radius divided by the packing radius. Among lattices, its minimum $γ_d$ is known for $d\leq 5$, attained by $\mathsf{A}_2^*$, $\mathsf{A}_3^*$, and Horváth's lattices $\mathsf{Ho}_4$, $\mathsf{Ho}