Batyrev–Manin's conjecture for big divisors

Let $X$ be a smooth projective variety over a number field $K$, with Néron–Severi space $\mathrm{NS}(X)_{\mathbb{R}}$ and pseudo-effective cone $\mathrm{PEf}(X)$. Assume $K_X\notin\mathrm{PEf}(X)$, and let $D$ be a big divisor. Define $\alpha(D)$ by

$$\alpha(D)[D]+[K_X]\in\partial\mathrm{PEf}(X),$$

and let $\beta(D)$ be the codimension of the minimal face of $\mathrm{PEf}(X)$ containing this boundary point. Batyrev–Manin's conjecture. If $K$ is sufficiently large, then for a sufficiently small cothin subset $U\subset X(K)$, with an arbitrary adelic metric on $D$,

$$\mathcal{N}_{U,D}(B)\sim CB^{\alpha(D)}(\log B)^{\beta(D)-1}.$$

This generalizes the predicted height asymptotic for Fano varieties and recovers the usual form when $D=-K_X$; the source gives no resolution status.