Batyrev--Manin--Peyre--Tschinkel conjecture for rational points

Let $\kappa$ be a number field and let $X$ be a smooth Fano variety over $\kappa$. Let $\mathcal{L}=(L,\{\|\cdot\|_v\})$ be an adelically metrized big and nef $\mathbb{Q}$-divisor, let $a(X,L)$ be the Fujita invariant, and let $b(\kappa,X,L)$ be the $b$-invariant. A subset of $X(\kappa)$ is thin if it is contained in a finite union of images of thin maps. Batyrev--Manin--Peyre--Tschinkel conjecture. If $X(\kappa)$ is not thin, then there exists a thin set $Z\subset X(\kappa)$ such that

$$N(X(\kappa)\setminus Z,\mathcal{L},T)\sim c(\kappa,\mathcal{L},Z)T^{a(X,L)}(\log T)^{b(\kappa,X,L)-1}$$

as $T\to\infty$.

The conjecture predicts the asymptotic growth of rational points after removing an exceptional thin set; the leading constant is Peyre's constant. Its general validity remains open.