Specialness criterion for standard linear systems in projective three-space

Let $\mathcal L=\mathcal L_3(d,m_1,\ldots,m_r)$ be a linear system in standard form. Let $Q=\mathcal L_3(2,1^9)$ be a quadric and let $\ell=\ell_3(1,1^2)$ be a line. The products $Q(\mathcal L-Q)(\mathcal L-K)$ and $\mathcal L\ell$ are intersection products, with $K$ denoting the canonical class.

Specialness criterion. The system $\mathcal L$ is special if and only if at least one of the following holds:

1. There exists a quadric $Q$ such that

$$Q(\mathcal L-Q)(\mathcal L-K)<0.$$

2. There exists a line $\ell$ such that

$$\mathcal L\ell\leq -2.$$

This conjecture is intended to provide a procedure for computing the dimension of a linear system: Cremona transformations and removal of fixed components reduce the problem to systems in standard form, whose speciality would then be detected by these two conditions.