Finiteness of the relative kernel group for Calabi–Yau fibrations

Let $f:(X,\Delta)\to S$ be a klt Calabi–Yau fibration such that $R^1f_*\operatorname{O}_X$ is a torsion sheaf. Let $\Gamma_B$ be the image of $\operatorname{PsAut}(X/S,\Delta)$ in $\operatorname{GL}(N^1(X/S)_\mathbb{R})$, and let $W$ be the maximal vector space contained in the relative movable cone $\overline{\operatorname{Mov}}(X/S)$. Define

$$\Gamma_W=\{\gamma\in\Gamma_B\mid \gamma\text{ acts trivially on }N^1(X/S)_\mathbb{R}/W\}.$$

Finiteness conjecture. The group $\Gamma_W$ is always finite. The preceding lemma proves finiteness under the stronger assumption that $X$ has terminal singularities; the conjecture asks for the stated klt generality.