Characterization conjecture for normal four-dimensional cyclic polytopes

Let $C_4(\tau_1,\tau_2,\dotsc,\tau_n)$ be a cyclic polytope, and write $\Delta_{ij}=\tau_j-\tau_i$ for $i<j$. A lattice polytope is normal when every lattice point in every positive dilate is a sum of lattice points of the polytope. Normality characterization conjecture. A cyclic polytope of dimension $4$ is normal if and only if

$$\Delta_{23}\geq 2\quad\text{and}\quad\Delta_{n-2,n-1}\geq 2.$$

The source motivates this as a complete characterization based on computational evidence and a preceding non-normality proposition; its resolution is not specified.