Degree-jumping conic containment conjecture

Let $Z$ be a collection of points, and let $\alpha_{a,b}(Z)$ denote the degree-jumping invariant used in the paper. Assume that $k\geqslant 5$ and that

$$\alpha_{k,k-1}(Z)=\alpha_{k-1,k-2}(Z)=\alpha_{k-2,k-3}(Z)=\alpha_{k-3,k-4}(Z)=2.$$

Degree-jumping conic containment conjecture. Then $Z$ is contained in a single conic.

This conjecture concerns the geometric consequences of four consecutive degree jumps by $2$. The supplied context does not establish a resolution, so the conjecture is recorded as open; the precise definition of $\alpha_{a,b}(Z)$ should be checked in the source.