The SI-sequence conjecture for smooth ACM curves

Let $X\subset\mathbb{P}^3$ be a smooth arithmetically Cohen–Macaulay curve not lying on a quadric surface. Let $AV_{X,1}$ denote its associated $AV$-sequence, and let $\deg X$ be the degree of $X$.

SI-sequence conjecture. The sequence $AV_{X,1}$ is an SI-sequence shifted by $1$. Its last nonzero term is

$$AV_{X,1}(\deg X-5),$$

so the SI-sequence ends in degree $\deg X-6$.

The statement is presented as partly conjectural and supported by experiments and partial results. The supplied text does not provide a resolution.