Campana's conjecture over function fields

Let $X$ be a smooth projective variety over $\a9\mathbb{C}$ and let $D$ be a strict normal crossings divisor on $X$ with irreducible components $D_1,\ldots,D_q$. For each $1\leq j\leq q$, let $\a9\epsilon_j>0$ be rational and set $\overline{\epsilon}=(\epsilon_1,\ldots,\epsilon_q)$ and $D_{\overline{\epsilon}}=\sum_{j=1}^q\epsilon_jD_j$. Let $K_X$ be a canonical divisor, let $C$ be an $(X,D_{\overline{\epsilon}})$-Campana curve with normalization $\nu_C:\widetilde C\to X$, let $g(C)$ denote the genus of $\widetilde C$, and let $h_{\mathcal A}(\nu_C)$ be the height associated with an ample sheaf $\mathcal A$. If

$$K_X+(D-D_{\overline{\epsilon}})$$

is big, then Campana's conjecture. There exist a proper Zariski closed subset $Z_{\overline{\epsilon},\omega,D}\subseteq X$, an ample sheaf $\mathcal A$ on $X$, and a constant $B$ depending on the preceding data, in particular on $\overline{\epsilon}$ and $D$, such that every $(X,D_{\overline{\epsilon}})$-Campana curve $C\subseteq X$ not contained in $Z_{\overline{\epsilon},\omega,D}$ satisfies

$$h_{\mathcal A}(\nu_C)\leq B\max\{0,g(C)-1\}.$$

This is the function-field form of Campana's conjecture, predicting height control—and hence algebraic degeneracy outside a proper exceptional set—for Campana curves when the orbifold canonical divisor is big. The source presents it as a conjectural statement and refers to the analogous number-field formulation and to existing partial results, so its resolution status is not established here.