A Manin-type conjecture for Campana Fano orbifolds

Let $(X,D_\epsilon)$ be a Campana orbifold over a number field $F$ such that $-(K_X+D_\epsilon)$ is ample, and let $(\mathcal X,\mathcal D_\epsilon)$ be a good integral model over the ring of $S$-integers $\mathcal O_{F,S}$, where $S$ contains all archimedean places. Assume $\lfloor D_\epsilon\rfloor=0$, so the orbifold is klt, and let $\mathsf H_{\mathcal L}$ be the height associated with an adelically metrized big line bundle $\mathcal L=(L,\|\cdot\|)$, with counting function

$$\mathsf N(U,\mathcal L,T)=\#\{P\in U\mid \mathsf H_{\mathcal L}(P)\leq T\}.$$

Suppose that $L$ is nef and that the set of klt Campana points $(\mathcal X,\mathcal D_\epsilon)(\mathcal O_{F,S})$ is not thin. Manin-type conjecture for Fano orbifolds. There exists a thin set $Z\subset(\mathcal X,\mathcal D_\epsilon)(\mathcal O_{F,S})$ such that

$$\mathsf N\bigl((\mathcal X,\mathcal D_\epsilon)(\mathcal O_{F,S})\setminus Z,\mathcal L,T\bigr)\sim c(F,S,(\mathcal X,\mathcal D_\epsilon),\mathcal L,Z)T^{a((X,D_\epsilon),L)}(\log T)^{b(F,(X,D_\epsilon),L)-1}$$

as $T\to\infty$, where

$$a((X,D_\epsilon),L)=\inf\{t\in\mathbb R\mid tL+K_X+D_\epsilon\in\operatorname{Eff}^1(X)$$

is the Fujita invariant, and $b(F,(X,D_\epsilon),L)$ is the codimension of the minimal supported face of $\operatorname{Eff}^1(X)$ containing $a((X,D_\epsilon),L)[L]+[K_X+D_\epsilon]$. The constant $c(F,S,(\mathcal X,\mathcal D_\epsilon),\mathcal L,Z)$ is a positive Tamagawa constant.