(-1)-speciality conjecture for linear systems on rational surfaces

Let $S$ be a rational surface, let $D$ be an ample, non-special divisor on $S$, and let $p_1,\ldots,p_r$ be $r$ points on $S$. Write

$${\mathcal L}={\mathcal L}(D,m_1,\ldots,m_r)$$

for the corresponding linear system, and call it $(-1)$-special if the procedure of subtracting rational curves $C$ with negative intersection ${\mathcal L}\cdot C\leq -1$ produces a system whose virtual dimension is larger than that of the initial system. The $(-1)$-speciality conjecture. The linear system ${\mathcal L}$ is special if and only if it is $(-1)$-special. This conjecture proposes a criterion for speciality of linear systems on rational surfaces: the known examples motivate detecting speciality through the prescribed subtraction procedure, while the equivalence in this generality remains open.