Proper constructibility conjecture for the exceptional set

Let $\pi:\mathcal X\to B$ be a good Fano fibration and let $\mathsf Z\subset\mathcal X_{\eta}(K(B))$ be the exceptional set. A properly constructible thin set is a finite union of images of generic-fiber rational points under finite thin $B$-maps from geometrically integral projective $B$-varieties. Proper constructibility conjecture for the exceptional set. The exceptional set $\mathsf Z$ is a properly constructible thin set. This conjecture is the finiteness input for removing exceptional loci in the all-height counting problem; the source gives no resolution status.