Manin's conjecture for rational points on singular Del Pezzo surfaces

Let $S$ be a Del Pezzo surface over $\mathbb{Q}$ with at most rational double points, and let $S^\circ\subset S$ be a dense open subset in the Zariski topology. Let $-K_S$ denote the anticanonical divisor, let $N(S^\circ,-K_S,B)$ count rational points of $S^\circ$ of anticanonical height at most $B$, and let $r$ be the rank of the Picard group of the minimal desingularization $\widetilde S$ of $S$ over $\mathbb{Q}$. Manin's conjecture. There exists such an $S^\circ$ for which

$$N(S^\circ,-K_S,B)\sim c_{S,H}\cdot B(\log B)^{r-1}$$

as $B\to\infty$. This is the expected asymptotic formula for rational points on Del Pezzo surfaces, with a positive constant determined by the surface and the height; the statement is presented as a conjecture in the source, and its general validity remains open.