Structure conjecture for linear systems on generic K3 surfaces

Let $S$ and $\mathcal L$ be as above: $S$ is a generic $K3$ surface with ${\rm Pic}(S)=\mathbb{Z}H$, $H^2=2g-2\geq 2$, and $\mathcal L=\mathcal L^n(d,m_1,\ldots,m_r)$ is the system of curves in $|dH|$ through general points $p_i$ with multiplicities $m_i$, where $n=2g-2$. A fixed component is an irreducible curve contained in every divisor of the system. Structure conjecture. (i) $\mathcal L$ is special if and only if $\mathcal L=\mathcal L^4(d,2d)$ or $\mathcal L=\mathcal L^2(d,d^2)$ with $d\geq 2$; (ii) if $\mathcal L$ is non-empty, then its general divisor has exactly the imposed multiplicities at the points $p_i$; (iii) if $\mathcal L$ is non-special and has a fixed irreducible component $C$, then either $\mathcal L=\mathcal L^2(m+1,m+1,m)=mC+\mathcal L^2(1,1)$ with $C=\mathcal L^2(1,1^2)$, or $\mathcal L=2C$ with $C\in\{\mathcal L^4(1,1^3),\mathcal L^6(1,2,1),\mathcal L^{10}(1,3)\}$, or $\mathcal L=C$; (iv) if $\mathcal L$ has no fixed components, then either its general element is irreducible or $\mathcal L=\mathcal L^2(2,2)$. This conjecture is described as a translation to $K3$ surfaces of the Gimigliano–Harbourne–Hirschowitz conjecture. The paper says that it proves this conjecture equivalent to the Segre-like conjecture above, but the supplied source does not establish either conjecture in full.