The stronger bounded negativity conjecture for blow-ups of the plane

Let $\operatorname{Bl}_{n}\mathbb{P}^2$ be the blow-up of $\mathbb{P}^2$ at $n$ very general points, and let $C$ be an integral curve in this surface. A $(-1)$-curve means a curve with the usual $(-1)$-curve property.

Stronger bounded negativity conjecture. If $C$ is an integral curve in $\operatorname{Bl}_{n}\mathbb{P}^2$, then

$$C^2\geq -1$$

and $C^2=-1$ implies that $C$ is a $(-1)$-curve.

The source presents this as a stronger form of bounded negativity for blow-ups of $\mathbb{P}^2$ at very general points. The supplied text gives no resolution status.