Thinness conjecture for the exceptional set

Let $\pi:\mathcal{X}\to B$ be a good Fano fibration, let $\mathcal{X}_{\eta}$ be its generic fiber, and let $\mathsf Z$ be the union of the images of all breaking thin maps into $\mathcal{X}_{\eta}$. Thinness conjecture for the exceptional set. The exceptional set $\mathsf Z\subset\mathcal{X}_{\eta}(K(B))$ is a thin subset of $\mathcal{X}_{\eta}(K(B))$. The source states that this was proved in characteristic zero by LST18, Theorem 5.7; its validity in the characteristic-$p$ setting considered here remains the relevant issue.