Nagata–Iarrobino conjecture for generic points in affine space

Let $K$ be a field and let $p_1,p_r\in\mathbb{A}^{n}$ be generic points, meaning that their coordinates are algebraically independent over the prime field of $K$. Let $I$ be the ideal of these points, and let $\gamma(I)$ denote the Waldschmidt constant

$$\gamma(I)=\lim_{m\to\infty}\frac{\alpha(I^{m})}{m}.$$

Nagata–Iarrobino conjecture. If $I$ is the ideal of $r\gg0$ generic points of $\mathbb{A}^{n}$, then

$$\gamma(I)=\sqrt[n]{r}.$$

Nagata formulated this for $n=2$, while Iarrobino proposed the higher-dimensional version. The problem is open for $n>1$ and sufficiently large $r$ in the source.