The list conjecture for negative curves on blown-up surfaces

Let $X=\operatorname{Bl}_r Y$ be the blow-up of a surface $Y$ at $r$ points. Let $C\subset X$ be a nonexceptional integral curve with $C^2<0$. List conjecture. There exist a positive number $\nu=\nu_X$ and a nonnegative integer $\pi=\pi_X$ such that $C$ is a $(-n,p)$-curve for some $1\leqslant n\leqslant \nu$ and $0\leqslant p\leqslant \pi$; equivalently, the possible negative curves belong to a finite list of types. This conjecture proposes simultaneous bounds on the parameters of negative curves and is presented as a consequence of the expected boundedness of negativity.