Helmke's conjecture on local global generation of adjoint line bundles

Work over an algebraically closed field $k$ of characteristic zero. Let $X$ be a smooth projective variety of dimension $n$, let $x\in X$ be a point, and let $L$ be an ample line bundle. Assume that

$$L^n>n^n$$

and that

$$L^d\cdot Z\geq n^d$$

for every subvariety $Z\subset X$ containing $x$, where $d=\dim Z<n$. Helmke's conjecture. The line bundle $\mathcal{O}_X(K_X+mL)$ is globally generated at $x$. This stronger conjecture is motivated by results of Reider, Ein and Lazarsfeld, Fujita, Helmke, and others on effective bounds for adjoint linear systems. The source gives no resolution of this conjecture.