Iitaka fibration conjecture for integral points on polarized log Fano varieties

Let $\mathcal{X}=(\mathfrak{X},D,L)$ be a polarized log Fano variety, with integral-point scheme $\mathfrak{U}=\mathfrak{X}\setminus\overline{D}$, height function $H_L$, and counting function

$$N_{\mathcal{X},Z}(B)=\#\{x\in\mathfrak{U}(\mathds{Z})\setminus Z\mid H_L(x)\leq B\}.$$

Assume that $\mathfrak{U}(\mathds{Z})$ is not thin. Let $\phi=\phi_{(X,L)}\colon X\dashrightarrow W$ be the Iitaka fibration induced by the adjoint bundle $E_{\mathcal{X}}=K_X+D+a(\mathcal{X})L$. For each $t\in W(\mathds{Q})$ for which the fiber is defined, let $X_t$ and $\mathfrak{X}_t$ be its closures in $X$ and $\mathfrak{X}$, and set $\mathcal{X}_t=(\mathfrak{X}_t,D\cap X_t,L|_{X_t})$. A fiber is adjoint rigid when its adjoint bundle is rigid. Iitaka fibration conjecture. There are thin sets $Z\subset X(\mathds{Q})$ and $Y\subset W(\mathds{Q})$ such that for every $t\in W(\mathds{Q})\setminus Y$, the fiber $\mathcal{X}_t$ is adjoint rigid with $a(\mathcal{X}_t)=a(\mathcal{X})$ and $b(\mathcal{X}_t)=b(\mathcal{X})$, the sum

$$c(\mathcal{X})=\sum_{t\in W(\mathds{Q})\setminus Y}c(\mathcal{X}_t)$$

converges, and

$$N_{\mathcal{X},Z}(B)\sim c(\mathcal{X})B^{a(\mathcal{X})}(\log B)^{b(\mathcal{X})-1}.$$

This conjecture proposes that, in the non-adjoint-rigid case, the leading constant and asymptotic count decompose through the Iitaka fibration into contributions from adjoint-rigid fibers. The supplied text does not give evidence resolving the conjecture.