Strong Fujita freeness conjecture

Let $X$ be a normal projective variety of dimension $n$, let $x_0\in X$ be a smooth point, and let $L$ be an ample Cartier divisor. Assume that there exist positive numbers $\sigma_p$ for $p=1,2,\ldots,n$ such that, for every subvariety $W$ of dimension $p$ containing $x_0$,

$$\sqrt[p]{L^p\cdot W}\geq \sigma_p,$$

and such that

$$\sigma_p\geq n \quad\text{for all }p, \qquad \sigma_n>n.$$

Strong Fujita freeness conjecture. The linear system

$$|K_X+L|$$

is free at $x_0$.

The paper describes this as a strong version of Fujita's freeness conjecture, strengthening the conclusion by imposing local numerical positivity conditions on subvarieties through a smooth point. Its status is not specified in the supplied text.