Pieropan–Smeets–Tanimoto–Várilly-Alvarado counting conjecture for Campana points

Let $K$ be a number field, let $(X,D_{\epsilon})$ be a smooth klt Fano Campana orbifold over $K$, and let $(\mathcal{X},\mathcal{D}_{\epsilon})$ be a regular $\mathcal{O}_{K,S}$-model for a finite set $S$ of places containing $S_{\infty}$. Let $\mathcal{L}=(L,\|\cdot\|)$ be an adelically metrised big line bundle with height function $H_{\mathcal{L}}$, and write

$$N(U,\mathcal{L},B)=\#\{P\in U:H_{\mathcal{L}}(P)\leq B\}.$$

For a variety $V$ over a characteristic-zero field and a subset $A\subset V(k)$, recall that $A$ is thin if it is contained in a finite union of subsets of types I and II, where type I means containment in the $k$-points of a proper Zariski closed subset and type II means containment in the image of the $k$-points of a finite surjective morphism of degree at least $2$ from a normal geometrically irreducible variety of the same dimension. Pieropan–Smeets–Tanimoto–Várilly-Alvarado conjecture. Suppose that $L$ is nef and $(\mathcal{X},\mathcal{D}_{\epsilon})(\mathcal{O}_{K,S})$ is not thin. Then there exists a thin set $Z\subset (\mathcal{X},\mathcal{D}_{\epsilon})(\mathcal{O}_{K,S})$ and explicit positive constants $a=a((X,D_{\epsilon}),L)$, $b=b(K,(X,D_{\epsilon}),L)$, and $c=c(K,S,(\mathcal{X},\mathcal{D}_{\epsilon}),\mathcal{L},Z)$ such that, as $B\to\infty$,

$$N\bigl((\mathcal{X},\mathcal{D}_{\epsilon})(\mathcal{O}_{K,S})\setminus Z,\mathcal{L},B\bigr)\sim cB^a(\log B)^{b-1}.$$

This is a Manin-type asymptotic for Campana points, predicting polynomial-logarithmic growth after removing a thin exceptional set. The supplied source gives no evidence resolving the conjecture, so its status remains open.