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We prove the unimodality of the Ehrhart $\delta$-polynomial of the chainpolytope of the zig-zag poset, which was conjectured by Kirillov. First, basedon 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, itsunimodality immediately follows from a result of Gasharov, which states thatthe $W$-polynomials of naturally labeled graded posets of rank $1$ or $2$ areunimodal.

The unimodality of the Ehrhart $\delta$-polynomial of the chain polytope of the zig-zag poset