The strict inequality conjecture for the first two saturation indices

For each integer $d\geqslant6$, let $\alpha_1$ and $\alpha_2$ denote the first two entries of the saturation sequence of the rational normal curve of degree $d$. Strict inequality conjecture. For all $d\geqslant6$, one has

$$\alpha_1>\alpha_2.$$

The source states that this conjecture arises from the preceding computational data and that the author has been unable to make progress on it. Its status therefore remains open.