Effective nonvanishing for weighted complete intersections of codimension two

We show Kawamata's effective nonvanishing conjecture (also known as the Ambro--Kawamata nonvanishing conjecture) holds for quasismooth weighted complete intersections of codimension $2$. Namely, for a quasismooth weighted complete intersection $X$ of codimension $2$ and an ample Cartier divisor $H$ on $X$ such that $H-K_X$ is ample, the linear system $|H|$ is nonempty.