Fano fourfold of index $2$. However the same proof works in our setting as well. We reproduce the proof here for the convenience of the reader.

\[lem:3\] Let $X \to Y$ be a contraction of a $K_X$-negative extremal face. If there is a $3$-dimensional fiber, then $X$ is a ${{\mathbb}{P}}^1$ bundle over a smooth projective Fano $3$-fold.

Let $D$ be a $3$-dimensional fiber. There is an extremal ray $R$, which does not lie in the extremal face corresponding to the contraction $X \to Y$ and spanned by the class of a rational curve $C$ such that $C \cdot D >0$. Then the corresponding contraction of $R$ have fiber dimension at most $1$. Indeed if there is a $2$-dimensional fiber $E$, then $D \cap E$ is non-empty and has a $1$-dimensional component. But the class of this component has to lie in both $R$ and the extremal face, which is impossible. Then the contraction corresponding to $R$ is either a blow-up along a smooth surface or a conic bundle over a smooth $3$-fold by [@Ando]. The former case is impossible because of the assumption on the pseudo-index. For the same reason, the conic bundle is a ${{\mathbb}{P}}^1$-bundle. The base $3$-fold is 