Segre-like conjecture for linear systems on generic K3 surfaces

Let $S$) be a generic $K3$ surface with ${\rm Pic}(S)=\mathbb{Z}H$, where $H^2=2g-2\geq 2$. Let $p_1,\ldots,p_r$ be points in general position on $S$, with prescribed multiplicities $m_1,\ldots,m_r$, and let

$$\mathcal L=\mathcal L^n(d,m_1,\ldots,m_r),\qquad n=2g-2,$$

be the linear system of curves in $|dH|$ having multiplicity $m_i$ at $p_i$. A linear system is special when its imposed conditions are dependent, equivalently when its first cohomology does not vanish. Segre-like conjecture. If $\mathcal L$ is non-empty and reduced, then it is non-special. This is presented as a Segre-type prediction for systems through general fat points on generic $K3$ surfaces; the paper states that no previous conjecture concerning their structure had been formulated and later proves that this conjecture is equivalent to the more detailed conjecture in the paper. Its status is not resolved in the supplied source.