 $$\kappa(X,L)=\max\{\kappa(X,N_m)\mid m\in\mathcal S\}\geq0.$$ Pick a rational number $0<\varepsilon\ll0$ such that the pair $(X,\varepsilon L)$ is klt. Then $\varepsilon L\sim_\Q K_X+\varepsilon L$ is semiample