The cone-of-curves formulation of the negative-curve conjecture

Let $Y$ be the blow-up of $\mathbb{P}^2$ at a set of $r$ points in very general position. Write

$$\partial \overline{\operatorname{NE}}(Y)=\mathcal{B}_1\sqcup\mathcal{B}_2$$

as in the source, where $\mathcal{B}_1$ is the closure of the union of the facets of the boundary of the effective cone and $\mathcal{B}_2$ is supported by $\operatorname{Nul}(Y)$. Let $K_Y^{\leq 0}$ and $K_Y^{>0}$ denote the corresponding half-spaces, and let $\operatorname{Pos}(Y)$ and $\operatorname{Nul}(Y)$ be the positive cone and its boundary.

Cone-of-curves formulation. The extremal rays of $\mathcal{B}_1$ that do not lie on $\operatorname{Nul}(Y)$ are spanned by classes of $(-1)$-curves, and

$$\mathcal{B}_1=\partial\overline{\operatorname{NE}}(Y)\cap K_Y^{\leq 0},\qquad \mathcal{B}_2=\partial\overline{\operatorname{NE}}(Y)\cap K_Y^{>0}.$$

In particular,

$$\overline{\operatorname{NE}}(Y)\cap K_Y^{\geq 0}=\operatorname{Pos}(Y)\cap K_Y^{\geq 0}.$$

This is presented as a more precise formulation of the negative-curve conjecture and describes the boundary of the effective cone for very general blow-ups of $\mathbb{P}^2$. The source does not give evidence that this formulation has been solved.