TITLE: Voronoi cell of lattices with the same profile
QUESTION [22 upvotes]: Definition 1. Given a body $V$ in $\mathbb R^n$,
the function $p_V\colon \mathbb R_+\to \mathbb R_+$
$$p_V(r)=\mathop{\rm vol} [V\cap B_r(0)]$$
will be called profile of $V$.
Definition 2. Define Voronoi cell of lattice $L$ in $\mathbb R^n$ as 
$$V_L=\{\,x\in \mathbb R^n;\,|x|\le |x+\ell| \ \text{for any}\ \ell\in L\,\}.$$ 

Question. Can it happen that Voronoi cells of a pair of lattices have the same profile, but not isometric?

Comment

The question is inspired by this one.

REPLY [4 votes]: My apologies for a trivial comment (not an answer), well-understood by the OP.
But perhaps
this illustration will inspire advancement on this long-unsolved question,
and invite clarification if I am misinterpreting.
I abandon the requirement that $V$ be the Voronoi cell of a lattice, and just
look at convex bodies:
   
Here the two shapes have the same "profile" in Anton's sense:
they intersect origin-centered balls in the same volumes.
But they are not congruent, not equivalent under an isometry.
So the challenge is to arrange something similar for lattice Voronoi cells.