Lattice-Like Total Perfect Codes

A contribution is made to the classification of lattice-like total perfect codes in integer lattices $Λ_n$ via pairs ($G, Φ$) formed by abelian groups $G$ and homomorphisms $Φ: Z^n → G$. A conjecture is posed that the cited contribution covers all possible cases. A related conjecture on the unfinished work on open problems on lattice-like perfect dominating sets in $Λ_n$ with induced components that are parallel paths of length > 1 is posed as well.