The eventual terminal-3 conjecture for saturation sequences

For each positive integer $d$, let $(\alpha_1,\ldots,\alpha_{e_d-1})$ be the saturation sequence of the rational normal curve of degree $d$. Terminal-3 conjecture. For every positive integer $s$, there exists an integer $N_s$ such that, for all $d\geqslant N_s$, at least the last $s$ entries of the saturation sequence are equal to $3$:

$$\alpha_{e_d-s}=\alpha_{e_d-s+1}=\cdots=\alpha_{e_d-1}=3.$$

The paper proves this phenomenon for $s\leqslant8$ by explicit determinant calculations, but the assertion for arbitrary positive integers $s$ remains open.