  We prove the unimodality of the Ehrhart $\delta$-polynomial of the chain
polytope of the zig-zag poset, which was conjectured by Kirillov. First, based
on a result due to Stanley, we show that this polynomial coincides with the
$W$-polynomial for the zig-zag poset with some natural labeling. Then, its
unimodality immediately follows from a result of Gasharov, which states that
the $W$-polynomials of naturally labeled graded posets of rank $1$ or $2$ are
unimodal.
