Rank-index conjecture for a curve union its trisecant line

Let $\mathcal{C}\subset\operatorname{\mathbb{P}}^r$, let $\mathbb{L}$ be its unique trisecant line, and set

$$X=\mathcal{C}\cup\mathbb{L}.$$

Assume these objects are as in the paper's second main theorem. The rank index $\operatorname{rank-index}(X)$ is the invariant studied there. Rank-index conjecture.

$$\operatorname{rank-index}(X)=3\quad\text{if and only if}\quad\mathcal{C}\cap\mathbb{L}\text{ consists of three simple points}.$$

The preceding theorem establishes partial results, including rank index $4$ when the intersection has at most two points; the case of three simple points is the remaining situation addressed by this conjecture.