The monotonicity conjecture for the saturation sequence of the rational normal curve

For each positive integer $d$, let $(\alpha_1,\alpha_2,\ldots,\alpha_{e_d-1})$ be the saturation sequence, and let

$$\mathfrak S(d)=\max\{\alpha_1,\alpha_2,\ldots,\alpha_{e_d-1}\}.$$

Monotonicity conjecture. The saturation sequence is non-increasing:

$$\alpha_1\geqslant\alpha_2\geqslant\cdots\geqslant\alpha_{e_d-1}.$$

If true, this would imply that $\mathfrak S(d)=\alpha_1$. The conjecture is presented as an open question, and no progress is reported in the source.