Cohomological special effect conjecture for hypersurface linear systems

Let ${\mathcal{L}}:={\mathcal{L}}_{n,d}(-\sum_{i=1}^hm_iP_i)$ be a system of hypersurfaces in $\mathbb{P}^n$ with general multiple base points. A variety $Y$ is an $h^1$-special effect variety for ${\mathcal{L}}$ when it satisfies the source's conditions, including $h^0({\mathcal{L}}_{|Y})=0$, $h^0({\mathcal{L}}-Y)\ne0$, and $h^1({\mathcal{L}}_{|Y})>h^2({\mathcal{L}}-Y)$. A system is cohomologically special when it arises from such an $h^1$-special effect curve.

Cohomological special effect conjecture. The system ${\mathcal{L}}$ is special if and only if it is cohomologically special.

The conjecture extends the cohomological special-effect criterion to higher-dimensional hypersurface systems. The supplied text gives no resolution evidence.