Existence of isolated n-PTMCs in Γn

Let $\Gamma_n$ denote the graph considered in the paper, and let an isolated $n$-PTMC mean an isolated perfect total-multipacking code with parameter $n$. Existence conjecture. There exists an isolated $n$-PTMC in $\Gamma_n$ for every $n>2$. The supplied context describes an iterative construction for the case $n=2$ and suggests the higher-dimensional existence claim, but does not provide a resolution for all $n>2$.