Suppose $ \pi: X \dottedrightarrow & Y$ is a birational map which is an isomorphism in codimension 1 (such as a flip). Also suppose both $X$ and $Y$ have reasonable (say log terminal) singularities. We know that divisors in $X$ and $Y$ correspond bijectively to each other. If $D$ is a nef divisor in $X$, is it true that $\overline{\pi(D)}$ is also nef in $Y$?

Suppose $\pi: X \dashrightarrow Y$ is a birational map which is an isomorphism in codimension 1 (such as a flip). Also suppose both $X$ and $Y$ have reasonable (say log terminal) singularities. We know that divisors in $X$ and $Y$ correspond bijectively to each other. If $D$ is a nef divisor in $X$, is it true that $\overline{\pi(D)}$ is also nef in $Y$?