Batyrev–Tschinkel's modified Manin conjecture for canonical Fano varieties

Let $X$ be a canonical Fano variety over a number field $K$, with a metrized anti-canonical divisor and associated height function $H$. For a suitable subset $U\subset X(K)$, write

$$N_U(B)=\sharp\{x\in U\mid H(x)\le B\}.$$

Let $\rho(X)$ be the Picard number and let $\operatorname{cd}(X)$ be the number of crepant divisors over $X$. Batyrev–Tschinkel's modified Manin conjecture. There is a positive constant $c$ such that

$$N_U(B)\sim c\cdot B\cdot(\log B)^{\rho(X)+\operatorname{cd}(X)-1}\qquad(B\to\infty).$$

This modifies the expected point-counting asymptotic for smooth Fano varieties to account for canonical singularities. The source presents it as a modification of a conjecture of Batyrev and Tschinkel; its general validity is not established.