Weak SHGH conjecture for nef linear systems

Let $X_n$ be the blow-up of the projective plane at $n$ general points, and let $\mathcal{L}$ be a nef linear system, meaning that $\mathcal{L}\cdot\mathcal{L}'\geq0$ for every effective linear system $\mathcal{L}'$. Weak SHGH conjecture. The linear system $\mathcal{L}$ is not special. This is presented as a weak form of the SHGH conjecture and is open in general.