Birational invariance of condition star for dP3 fibrations

Let $X\to\mathbb{P}^1$ be a $dP_3$ fibration with nonsingular total space, and let $X'\to\mathbb{P}^1$ be a threefold Mori fibre space square birational to $X/\mathbb{P}^1$. Assume that $X/\mathbb{P}^1$ satisfies condition $(\ast)$:

$$-K_X\notin\operatorname{Int}\operatorname{\overline{NM}}^1(X).$$

Birational invariance conjecture for condition star. Then $X'/\mathbb{P}^1$ also satisfies condition $(\ast)$. This predicts that condition $(\ast)$ is preserved under square-birational changes of Mori fibre space model.