Rank-index conjecture for projected rational normal curves

Let $\widetilde{\mathcal{C}}$ be the rational normal curve and let $p\in\operatorname{\mathbb{P}}^{r+1}\setminus\widetilde{\mathcal{C}}^3$ be a projection center, with $\mathcal{C}=\pi_p(\widetilde{\mathcal{C}})\subset\operatorname{\mathbb{P}}^r$. The rank index $\operatorname{rank-index}(\mathcal{C})$ is the least integer governing the quadratic rank condition used in the paper. Rank-index conjecture. For $\mathcal{C}$ as in the main theorem,

$$\operatorname{rank-index}(\mathcal{C})=3.$$

The main theorem proves this when $p$ is a coordinate point, while the conjecture asserts that the value is always $3$; the paper notes that no example with rank index $4$ is known.