Nagata's conjecture for plane curves through very general points

Let $p_1,\dots,p_r$ be $r$ very general points in $\mathbb P^2$. For an integral curve $C\subset\mathbb P^2$, write $\operatorname{mult}_{p_i}(C)$ for its multiplicity at $p_i$. Equivalently, on the blow-up $X\to\mathbb P^2$ at the points $p_i$, let $H$ be the pull-back of a line and let $E_i$ be the exceptional divisors; then $dH-\sum_{i=1}^r m_iE_i$ denotes the corresponding linear system. Nagata's conjecture. For every integral curve $C\subset\mathbb P^2$, one has

$$\deg C>\frac{1}{\sqrt r}\sum_{i=1}^r\operatorname{mult}_{p_i}(C).$$

Equivalently, the linear system $|dH-\sum_{i=1}^r m_iE_i|$ contains no integral curve for positive integers $d$ and $m_i$ satisfying

$$\sqrt r\leq\frac{1}{d}\sum_{i=1}^r m_i.$$

Nagata's conjecture is a long-standing open problem concerning positivity and linear systems on blow-ups of $\mathbb P^2$ at very general points; it is closely related to questions about the positivity and generation of line bundles on these surfaces.