A projective normal and $\mathbb{Q}$-factorial variety $X$ is said to be log Fano if there exists and effective divisor $D$ on $X$ such $-K_X-D$ is ample and the pair $(X,D)$ is klt.

Now, let $f:X\dashrightarrow Y$ birational map which is an isomorphism in codimension one between two projective normal and $\mathbb{Q}$-factorial varieties. Assume that the strict transform of $-K_X$ on $Y$ is nef and big. Can we conclude that then $X$ is log Fano? I know that this holds if the strict transform of $-K_X$ on $Y$ is ample.

Thank you very much.