Numerical special effect conjecture for plane linear systems

Let ${\mathcal{L}}_{2,d}(-\sum_{i=1}^hm_iP_i)$ be a linear system of plane curves with general multiple base points. A special system is numerically special if it arises from an $\alpha$-special effect variety or an $(\alpha_1,\dots,\alpha_r)$-special effect configuration, as defined in the source.

Numerical special effect conjecture. The system ${\mathcal{L}}_{2,d}(-\sum_{i=1}^hm_iP_i)$ is special if and only if it is numerically special.

This conjecture asks whether every special plane system is explained by the numerical special-effect construction. The supplied text gives no resolution evidence.