Secant-obstruction conjecture for linear systems through n+3 points

Let ${\mathcal L}$ be a non-empty linear system in ${\mathbb P}^n$ with $n+3$ base points in general position, and let $C$ be the rational normal curve through those points. Let $\sigma_t$ denote the $t$-th secant variety of $C$, and let $\operatorname{\sigma ldim}({\mathcal L})$ be the secant linear expected dimension defined by incorporating the contributions of the relevant linear cycles and cones over these secant varieties. A system is special when its dimension exceeds its ordinary expected dimension. Secant-obstruction conjecture. The system ${\mathcal L}$ is special only if its base locus contains either linear cycles or cones over the secant varieties $\sigma_t$ of $C$, and

$$\dim({\mathcal L})=\operatorname{\sigma ldim}({\mathcal L}).$$

The paper states that this conjecture is proved for $n\leq 3$ and for some homogeneous families, while the general case remains open in the supplied text.