Suppose that $v=(v_1,\ldots, v_d)\in \mathbb{R}^d$ lies in the linear subspace $v_1+\cdots +v_d=0$, and moreover that the coordinates are pairwise distinct. The permutahedron \begin{equation} P(\mathcal{S}_d;v)=Conv(\mathcal{S}_d\cdot v) \end{equation} is the convex hull of $v$ under the symmetric group action on coordinates. It is a $(d-1)$-dimensional polytope.

Now consider the cyclic subgroup $C_d$ of $\mathcal{S}_d$ generated by the permutation $(123\cdots d)$ and consider the corresponding orbitope $P(C_d; v)=Conv(C_d\cdot v)$.

Question: Is $P(C_d;v)$ necessarily a $(d-1)$-dimensional simplex?