Lazarsfeld's conjecture on Seshadri constants of ample line bundles

Let $X$ be a projective manifold and let $L$ be an ample line bundle on $X$. For a very general point $x\in X$, write

$$\varepsilon(X,L;1)=\varepsilon(X,L;x),$$

the maximal value of the Seshadri constant of $L$ on $X$. Lazarsfeld's conjecture. One has

$$\varepsilon(X,L;1)\geq 1.$$

This conjecture predicts a uniform lower bound for the local positivity of ample line bundles in every dimension. The paper presents it as a general conjecture; no resolution is given here.