For integral polytopes, it is conjectured (T. Hibi), that if the $h^*$-vector is symmetric, then it is also unimodal (increasing, then non-decreasing).

A non-integral polytope do not, in general, have a polynomial Ehrhart function. However, if it does, we can compute the corresponding $h^*$-vector.

Now, are there explicit examples where the conjecture mentioned above is false, published somewhere? That is, that the conjecture does not extend to non-integral polytopes of this special kind above?

I stumbled upon such an example in my current research, and I am just wondering if it is worth mentioning. The polytope in question is the Gelfand-Tsetlin polytope, $GT(\lambda,w)$ for $\lambda=(2,2,2)$ and $w=(1,1,1,1,1,1)$.