Helmke's numerical Fujita freeness conjecture

Let $X$ be a smooth projective variety of dimension $\operatorname{dim}(X)=n$, and let $L$ be an ample divisor such that

$$(L^n)>n^n,$$

and, for every irreducible cycle $Z\subseteq X$ of dimension $\operatorname{dim}(Z)=d<n$,

$$(L^d\mathbin{\cdot} Z)\geq n^d.$$

Helmke's numerical Fujita freeness conjecture. Under these conditions, the adjoint divisor

$$K_X+L$$

is globally generated. This numerical version is intended to imply Fujita's freeness statement from numerical positivity conditions and is currently open in general.