Nagata's ampleness conjecture for blowups of the projective plane

Let $X$ be the blowup of $\mathbb{P}^2$ at $n$ very general points, with $H$ the pullback of a line and $E=\sum_{i=1}^n E_i$ the sum of the exceptional divisors. For a real number $t$, set

$$A_t=tH-E.$$

Nagata's conjecture. If $n\geq 10$ and $t>\sqrt{n}$, then $A_t$ is ample. In particular,

$$B:=A_{\sqrt{n}}=\sqrt{n}H-E$$

is nef. Nagata proved the conjecture when $n$ is a perfect square; the general case remains open, with partial results obtained by approximating $\sqrt n$ with values of $t$ for which $A_t$ is ample.