Nagata's Kähler packing conjecture for blow-ups of the projective plane

Let $\sigma:X\to\mathbb{CP}^2$ be the blow-up at points $x_1,\ldots,x_k$, with $l$ the Poincaré dual of a line and $e_q$ the Poincaré duals of the exceptional divisors. For $k\geq 9$, Nagata's Kähler conjecture. If the points $x_1,\ldots,x_k$ are chosen sufficiently general, then for every $\epsilon<\frac{1}{\sqrt{k}}$ there is a Kähler form representing the cohomology class

$$\sigma^*l-\epsilon\sum_{q=1}^k e_q.$$

This is the Kähler formulation of Nagata's conjecture, arising from symplectic packing problems for the projective plane. Its relation to the algebraic-geometric inequality in the second candidate makes the two statements equivalent formulations rather than distinct conjectures.