Twisted-fibration asymptotic conjecture for locally soluble fibres

Let $\pi:X\to\mathbb{P}^1$ be a standard fibration, and let $\theta:\mathbb{P}^1\to(\mathbb{P}^1,\partial_\pi)$ be a finite étale orbifold morphism that is a $G$-torsor under a finite étale group scheme $G$. Assume that $(\mathbb{P}^1,\partial_\pi)$ does not admit a finite étale orbifold morphism factoring through $\theta$. For each $v\in H^1(\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}),G)$, let $\theta_v:\mathbb{P}^1\to\mathbb{P}^1$ be the twist of $\theta$ by $v$, and let $\pi_v:X_v\to\mathbb{P}^1$ be the normalisation of the pullback of $\pi$ along $\theta_v$. Let $\partial_\pi$ be the orbifold boundary defined by the minimum multiplicities of the fibres. Twisted-fibration asymptotic conjecture. If $-(K_{\mathbb{P}^1}+\partial_\pi)$ is ample and $X(\mathbf{A}_\mathbb{Q})\neq\emptyset$, then there exists $c_\pi>0$ such that

$$N_{\text{loc}}(\pi,B)\sim c_\pi\frac{B^{2-\deg\partial_\pi}}{(\log B)^{\min_{v\in H^1(\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}),G)}\Delta(\pi_v)}}.$$

Here $\Delta(\pi_v)$ is the quantity defined in the source's equation (the definition is not included in the supplied excerpt). This conjecture proposes an exact asymptotic, refining the preceding upper-bound conjecture; the excerpt does not provide evidence of resolution.