Non-split fibre conjecture for families of varieties

Let $F$ be a number field, let $X$ satisfy the stated assumptions, and let $H$ be an anticanonical height function on $X$. Let $\pi:Y\to X$ be a non-singular proper faithfully flat morphism with geometrically integral generic fibre and reduced fibres over codimension-one points. Assume that $Y(F)$ is Zariski dense and that, outside a Zariski closed subset of $X$, all fibres satisfy the Hasse principle. Call a fibre non-split if it does not contain a geometrically integral open subscheme. Non-split fibre conjecture. There exists a codimension-one point $D\in X$ whose fibre $\pi^{-1}(D)$ is non-split if and only if, for every sufficiently small Zariski open subset $U\subset X$,

$$N(U,H,\pi,B)=o\bigl(N(U,H,B)\bigr).$$

The conjecture proposes that the quantitative scarcity of rational fibres is governed by non-split codimension-one fibres. The supplied text gives no resolution evidence.