Nonexistence of lattice t-PTMCs in the intermediate range

Let $\Lambda_n=\mathbb{Z}^n$ and let a lattice $t$-PTMC be a lattice perfect total-multipacking code with parameter $t$. Nonexistence conjecture. There is no lattice $t$-PTMC in $\Lambda_n$ for $1<t<n$. The question concerns the existence of lattice perfect total-multipacking codes beyond the known constructions for $t=1$; its resolution is not indicated in the supplied text.