Maximal covering density of the lattices $LD_n$ among covering maxima

For each $n\geq 6$, let $LD_n$ be the lattice defined in Dutour (2005), and let a covering maximum be a lattice whose covering density decreases under perturbation. The covering-density conjecture for $LD_n$. The lattice $LD_n$ has maximal covering density among all covering maxima. This is motivated by the identification of $LD_n$ as a covering maximum and by the theorem classifying the seven-dimensional perfect Delaunay polytopes; partial enumerations in dimensions $8$, $9$, and $10$ support the conjecture, but the claim is not stated as resolved.