  The Voronoi conjecture on parallelohedra claims that for every convex
polytope $P$ that tiles Euclidean $d$-dimensional space with translations there
exists a $d$-dimensional lattice such that $P$ and the Voronoi polytope of this
lattice are affinely equivalent. The Voronoi conjecture is still open for the
general case but it is known that some combinatorial restriction for the face
structure of $P$ ensure that the Voronoi conjecture holds for $P$. In this
paper we prove that if $P$ is the Voronoi polytope of one of the dual root
lattices $\mathsf{D}_d^*$, $\mathsf{E}_6^*$, $\mathsf{E}_7^*$ or
$\mathsf{E}_8^*=\mathsf{E}_8$ or their small perturbations, then every
parallelohedron combinatorially equivalent to $P$ in strong sense satisfies the
Voronoi conjecture.
