The height comparison conjecture for primitive fibers

Let $Y$ be the base of the ${\cal L}$-primitive fibration in the source, let $U\subset Y$ be a dense Zariski open subset, and let $V_y$ be the corresponding ${\cal L}$-primitive fiber for $y\in Y(F)\cap U$. Let ${\cal F}$ be a metrized ${\bf Q}$-invertible sheaf associated with the ${\bf Q}$-Cartier divisor $L_1^{-1}\otimes K_Y$, where $L_1$ is the tautological ample ${\bf Q}$-Cartier divisor on $Y$ defined by the graded ring ${\rm R}(V, {\cal L})$.

Height comparison conjecture. There exist a family of $v$-adic metrics on $K_Y$ and constants $c_2>c_1>0$ such that

$$c_1H_{\cal F}(y)\leq\tau_{\cal L}(V_y)\leq c_2H_{\cal F}(y)\qquad\forall y\in Y(F)\cap U.$$

This conjecture proposes that the adelic Tamagawa quantity of a primitive fiber is comparable to a height on the base, supporting the predicted counting asymptotics. The supplied text gives no resolution status.