  In a recent paper Garber, Gavrilyuk and Magazinov proposed a sufficient
combinatorial condition for a parallelohedron to be affinely Voronoi. We show
that this condition holds for all five-dimensional Voronoi parallelohedra.
Consequently, the Voronoi conjecture in $\mathbb R^5$ holds if and only if
every five-dimensional parallelohedron is combinatorially Voronoi. Here, by
saying that a parallelohedron $P$ is combinatorially Voronoi, we mean that the
tiling $\mathcal T(P)$ by translates of $P$ is combinatorially isomorphic to
some tiling $\mathcal T(P')$, where $P'$ is a Voronoi parallelohedron, and that
the isomorphism naturally induces a linear isomorphism of lattices $\Lambda(P)$
and $\Lambda(P')$.
  We also propose a new sufficient condition implying that a parallelohedron is
affinely Voronoi. The condition is based on the new notion of the Venkov
complex associated with a parallelohedron.
