Chudnovsky's conjecture on Waldschmidt constants

Let $S=\{p_1,\dots,p_r\}$ be distinct points in $\mathbb P^N$ over an algebraically closed field, and let $a(S,m)$ and $e(S)$ be defined by

$$e(S):=\lim_{m\to\infty}\frac{a(S,m)}{m},\qquad a(S,m):=\min\{d:h^0(X,L(d,m))>0\}.$$

Chudnovsky's conjecture, proved for $N=2$, asserts

$$\frac{a(S,1)+N-1}{N}\leqslant e(S).$$

This strengthens the Waldschmidt--Skoda bound and would give sharper lower estimates for asymptotic effectivity; it remains open in higher dimensions.