The normality conjecture for toric rings arising from cyclic polytopes

Let $Q$ be the affine semigroup associated with the cyclic polytope construction in the paper, and let $K[Q]$ be its semigroup algebra over the field $K$. Write $n$ and $d$ for the parameters of the construction, and let $\tau_1<\tau_2<\cdots<\tau_n$ be the associated parameters. Normality conjecture. The $K$-algebra $K[Q]$ is normal only in one of the following cases: $n=d+1$; or $d=1$ and

$$\tau_2-\tau_1=\tau_3-\tau_2=\cdots=\tau_n-\tau_{n-1}.$$

This conjecture seeks a complete characterization of normality beyond the cases already established, including the known characterization when $d=1$ and the evident case $n=d+1$; the normality question remains open in the stated generality.