Negativity conjecture for prime divisors on blow-ups of the plane

Let $X_n$ be the projective plane blown up at $n$ general points, and let $C$ be a prime divisor on $X_n$ with self-intersection $C^2$ and arithmetic genus $g$. Negativity conjecture. One has $C^2\geq-1$, and if $C^2=-1$ then $g=0$. This conjecture is implied by SHGH and in turn implies Nagata's conjecture; the source does not report a general resolution.