Batyrev–Tschinkel's prediction for constants in the anticanonical counting asymptotic

Let $X$ be a Fano variety over $\mathds{Q}$ with at worst log terminal singularities and a Zariski dense set of rational points. Let $H\colon X(\mathds{Q}) \to \mathds{R}_{>0}$ be an anticanonical height function, let $\pi\colon \widetilde{X} \to X$ be a desingularization, and set $L \coloneqq \pi^*(-K_X)$. Define

$$\mathfrak{a}\coloneqq \inf\{t\in \mathds{R}: t \cdot L + K_{\widetilde{X}}\text{ is effective}\}$$

and let $\mathfrak{b}$ be the codimension of the minimal face of the effective cone of $\widetilde{X}$ containing $\mathfrak{a} \cdot L + K_{\widetilde{X}}$. Let $\phi\colon X \dashrightarrow P$ be the $\mathcal{L}$-primitive fibration associated with $\mathfrak{a} \cdot L + K_{\widetilde{X}}$, and write $\mathfrak{c}_x$ for the expected leading constant in the counting asymptotic on the fiber over $x\in P$. Batyrev--Tschinkel's prediction. Let $\overline{H}$ be a height on $P$ relative to the line bundle $\mathcal{O}_P(-1) \otimes \omega_P$. Then there exist positive constants $c_1,c_2$ and an open subset $V \subseteq P$ such that for every $x \in V$ we have

$$c_1 \overline{H}(x) \le \mathfrak{c}_x \le c_2 \overline{H}(x).$$

This predicts that the leading constants in the fiberwise Manin asymptotics vary comparably to a height associated with $\mathcal{O}_P(-1) \otimes \omega_P$ on a dense open subset of the base. The statement is attributed to Batyrev and Tschinkel and is presented as a prediction in the source; its resolution is not established here.