Nagata–Szemberg conjecture on maximal Seshadri constants

Let $X$ be a smooth variety of dimension $n$ and let $L$ be an ample divisor on $X$. For very general points $p_1,\dots,p_r\in X$, write $\varepsilon(L,r)=\varepsilon(L,p_1,\dots,p_r)$ for the multipoint Seshadri constant. Nagata–Szemberg conjecture. There exists a number $r_0=r_0(X,L)$ such that, for every $r\ge r_0$,

$$\varepsilon(L,r)=\sqrt[n]{\frac{L^n}{r}}.$$

The displayed quantity is the general upper bound for the Seshadri constant of $r$ points. Szemberg proposed this as a generalization of Nagata's conjecture to arbitrary smooth varieties; the paper proves it for certain smooth surfaces with an ample divisor generating $NS(X)$ and with $L^2$ a square, but the general assertion remains open.