The negative-curve conjecture for very general blow-ups of the plane

Let $Y$to be the blow-up of $\mathbb{P}^2$ at a set of points in very general position, and let $C$ be an integral curve on $Y$ with negative self-intersection.

Negative-curve conjecture. The curve $C$ is a $(-1)$-curve of $Y$, meaning that it is a smooth rational curve with self-intersection $-1$.

This is a weaker form of the Segre–Harbourne–Gimigliano–Hirschowitz conjecture and concerns the cone of effective curves on very general blow-ups of $\mathbb{P}^2$. The source presents it as a conjecture and gives new evidence, but does not state that it has been resolved.