Monotonicity conjecture for normal cyclic polytopes

Let $C_d(\tau_1,\dotsc,\tau_n)$ and $C_d(\tau_1',\dotsc,\tau_n')$ be integral cyclic polytopes, and let $\Delta_{ij}$ denote the corresponding gaps used in the source. Assume that $C_d(\tau_1,\dotsc,\tau_n)$ is normal, meaning that lattice points in all positive dilates decompose as sums of lattice points of the polytope. Monotonicity conjecture. If $C_d(\tau_1,\dotsc,\tau_n)$ is normal and

$$\tau_j'-\tau_i'\geq\Delta_{ij}\qquad\text{for all }1\leq i<j\leq n,$$

then $C_d(\tau_1',\dotsc,\tau_n')$ is also normal. The statement is motivated by the preceding conjectures and a theorem in the paper, but no resolution is supplied.