Let P be the 2-dimensional alcoved polytope with b=[16,9,0,9,12,17], pictured below.
Let \( w = x_1 \). Then:
- \( h^{<1>}(z) = -10z^3 + 11z \)
- $\vdots$
- \( h^{<62>}(z) = -620z^3 + 158844z^2 + 80104z \)
- \( h^{<63>}(z) = 21z^3 + 166656z^2 + 83370z \)
All \( h^{<r>}(z) \) with \( 1 \le r \le 62 \) have mixed-sign coefficients, so at least \( 63 \) dilates are needed for weighted \( h^* \)-positivity.