Positive proportion conjecture for locally soluble fibres

Let $X$ satisfy the stated assumptions, let $H$ be an anticanonical height function, and let $\pi:Y\to X$ be a proper surjective almost smooth morphism with geometrically integral generic fibre. Assume that $Y(\mathbf{A}_F)\neq\emptyset$ and that every codimension-one fibre $\pi^{-1}(D)$, for $D\in X^{(1)}$, is split. Let $N_{\mathrm{loc}}(U,H,\pi,B)$ count points in $U(F)$ of height at most $B$ whose fibres have adelic points. Positive proportion conjecture. There exists an open subset $U\subset X$ such that

$$\lim_{B\to\infty}\frac{N_{\mathrm{loc}}(U,H,\pi,B)}{N(U,H,B)}>0.$$

This predicts a positive-density supply of everywhere locally soluble members when no codimension-one fibre is non-split. The supplied text gives no resolution evidence.